The good news is that brain freeze in this situation means that your brain is working perfectly. When it comes time to answer the questions of when and how to make a turn in the bumps, your brain is telling you that you it does not have an answer to either question and it's not going to make one up and give you a wrong answer. Clearly, your only problem is that you need some answers. Welcome to the club! If it makes you feel any better, there are a lot of pros in this "question" club. The bad news is that the better you get, the more unanswered questions you get.
Some people have learned to ski bumps via the traverse method you describe. We call it "shopping for a turn". By the way, you don't have to make an off balance turn when you hit the side of the run. You can stop and do a kick turn and start another traverse. Unless you're a ski god, this is a valuable tactical approach for anyone to adapt to bumps that are too steep, too large, too hard and/or hidden too much (e.g. by fog). Most people learning this way eventually find a bump in the middle of the traverse they can turn on and then slowly get used to making more turns and doing less traversing. But it's a frustrating solution when everyone else on the run is just zipping along and it can take a painfully long time to learn bump skiing this way.
There are several aspects to bump skiing that seem to borrow from the movie catch-22. One of them is that if you find a good line, that will help you make good turns and if you make good turns, that will help you find a good line. To break through this "which came first, chicken or egg" type dilemma we need to break the problem into a few different parts.
The first part is making short radius turns on the groomed runs. If you have not mastered this, brain freeze in the bumps is highly likely. When you make really short turns on groomed snow, you get some rebound action out of the skis that almost makes the groomed run work just like the bumps. This kind of makes sense when you think about it because a bump run is just a groomer that has been made bumpy by skiers making short turns in the same spots over and over again. When you truly own short turns on the groomers, you've got a lot less that you need to think about when you're in the bumps.
The second part, as you've guessed, is speed control. There are many different specific ways to control speed in the bumps. For discussion purposes today, I'm going to lump them into 4 categories:
1) turn shape
2) edge angle
3) absorption and extension
4) snow texture
Traversing is just one way of making a very wide turn. The more you travel across the slope, the slower your speed will be down the slope. Just like on a groomed run, if you finish your turn going uphill, then you will get free braking from gravity.
In general, the more you get your skis on edge, the braking power you are going to get from them. The two basic approaches here are either getting your skis turned across the fall line and "setting" your edges to get a hockey stop like check or aggressively rolling your skis onto edge into the bump surface to get them to turn quicker.
Absorption and extension means using your lower body to maintain snow contact. If you make contact with a bump with stiff legs you will absorb the impact the hard way. If you bend your knees on contact, you soften the blow by stretching out the length of time of the impact. You will also transfer more of your momentum into the bump versus bouncing off of it. This, my friend, is braking - aka speed control! Extension is mostly just resetting the length in your legs so you are ready to absorb the next bump, but it also lets you get your skis back onto the snow quicker so that they have more snow to work into turns. This part (absorption and extension) is what's new to most beginner bumpers.
The last category of speed control is snow texture. Ruts get formed from skiers pushing snow to the sides of the rut. The bottom of the rut is firm and fast. The sides of the ruts and the tops of the bottom collect all that pushed snow and are generally softer and slower. So the more you travel in the rut, the faster you go. The more you travel out of the rut, the slower you go.
When you've got a good enough understanding of these speed control categories to start putting them together, you are well on your way to having a brain that has plenty of good answers when you need one.
The third part is vision. Beginners tend to focus only on the bump at hand because they need their eyes to help guide their feet. With experience, your feet won't need any help to get past the bump at hand just like you don't need to look down at each footstep when you're hiking a trail in the woods. This will let your eyes start looking more ahead, gathering more information and giving your brain time to "package" that information. As you get more experience, you'll begin to transition from seeing "this bump, that bump and the next bump" to seeing "sets of bumps" with "trouble spots" in between. With experience, you'll easily recognize long bumps and short bumps, soft snow and hard snow, steep walls versus traversable ruts and obstacles like a downed munchkin hiding in a rut or the oft dreaded ROCKS! It does not matter what skill level your at, knowing what to look for makes a big difference in bump skiing. It start with looking at a single bump and seeing all of it's skiable parts versus just seeing the rut or places you don't want to go. If you focus on places you don't want to go, that's where you'll end up.
The fourth part is tactics. I teach that there are 3 basic tactical approaches to bump skiing: in the ruts, across the ruts or top to top. In reality very few bump skiers go top to top all the way through a bump run and most runs involve mixing and matching tactics. Very few skiers ski directly in the rut line because it is faster than the original line before the bump was made because the rut is firmer than the original snow surface was. What skiers do is ride the edges of the ruts to get just enough soft snow to turn in and slow down. As you get more experience you develop a tactics bag of tricks for dealing with trouble spots (e.g. making an extra turn to cut a long mogul in half instead of getting stuck in a long rut and end up going to fast). Although tactics is really just putting the other parts together, it's the thought process that says "when I see this, I'm going to do that" that eliminates brain freeze. When you know that certain things are going to work, it's much easier to pull the trigger and just do it. After that, getting better in the bumps is just a matter of going faster, going steeper, developing a different style, picking up a new trick, etc.
There are many different ways to learn bump skiing. I've just introduced one way to think about bump skiing by breaking it down into 4 parts: short turns, speed control, vision and tactics. This may be enough to help you have a break through and start enjoying the bumps instead of struggling with them. There are lots of other resources that can help your quest too. If you search Epic for "bumps" and "moguls", you'll find a ton of threads on the topic and references to books and videos that can help too. Taking a bump lesson from a pro can either be a godsend because of a few simple tips leading to a breakthrough or a total bummer because the message might be that you need to do a lot of work on your short turns on groomers before you're ready to tackle the bumps. But one thing an instructor can do for you is give you an honest evaluation of where you're at and help you develop a plan for getting to where you want to go.
In a beginner bump lesson, I will typically focus on developing one turn that can be used for speed control and be used as a building block for getting down the trail without resorting to traversing. In that lesson I will touch on pieces of the four parts and may introduce exercises to help develop those skills. Here I've just attempted to answer your question without getting into a specific plan, because for that I need to know what kind of bumps we have available to work on and have a better idea of what your strengths and weaknesses are. Let us know if this approach has been helpful.
Good luck. A ton of fun is waiting for you out there.