Your basic answer is that you won't. The job you're doing right now? That's the job they hired you for. It may not be the job you want, but it's the job they expect from you. Nothing's going to change by itself. Right now, you have three meaningful choices.
Accept it and move on. Adapt to your new reality. It doesn't sound like this is acceptable to you, and I sympathize with you on this one, but it is an option. The bright side of being overqualified for your current job is that at least you know that you're capable of it.
Attempt to turn your current job into the job you want to have. From the description you've given, this is unlikely to work. You might be able to convince your two executives to delegate more work to you or assign you interesting things to do, but you've already made at least a few pushes in that direction, and they haven't responded in a meaningful way. They haven't responded to that yet, which suggests that they're not inclined to.
If you do want to go with this path (and it's probably at a good idea to at least make the attempt) I would suggest the following. Continue working at the level you are for maybe another week or two - long enough to settle in, get more understanding of the company and how things work, and so forth. Spend your extra energy on trying to gain more of that understanding (while being careful not to be annoying). Then pick whichever of your executives is most likely to be approachable on the matter, and ask them for a moment of their time. Once they grant it, tell them straight-up that you want more to do. It'd be even better if you can come up with specific things that you could do that you think would be helpful to the executive or to the company as a whole. If he seems to be responding well, ask for any advice on how to broach the topic with the other one. The key here, though, is that you have to be utterly, almost painfully straightforward. Don't be accusative, but do make it utterly clear where you stand. It's far too easy to just shrug and move on when someone just asks if they can assist. This one might work.
For the matter of being asked to move heavy things with a damaged wrist, speak with HR and ask them how to handle that sort of a situation in the future. They're likely to have useful feedback. As far as getting out from under random drek tasks you don't want? You probably can't, unless you can somehow make yourself so indispensably useful that they don't want to spend your time on them. That seems unlikely starting from your current state, though.
- The third answer, if you've done all that, and the results have not been satisfactory, is to go looking for another job. This time, though, you know what you're looking for. You can mention it in the interviews. You can get a bit passionate about it. You want to work hard and be super-useful and super-productive and you have what it takes to make that happen. That's great! If you state that outright and forcefully (but not too forcefully) in your interviews, and make clear that you would not be happy at a place that expected you to be more laid-back, you'll get more interest from the sorts of jobs you do what, and you'll turn off folks like your current employers who you wouldn't want to work for anyway.
As a side note, you might want to consider looking at relatively young tech companies, where they often expect basically everyone to be high-intensity.