would that reveal my potential intentions to move my career
The employer would be astoundingly stupid to think that most of its employees don't have at least potential intentions to go elsewhere. Ongoing layoffs do that to your workforce. So you wouldn't really be "revealing" anything. You'd be confirming what they already think they know.
The reason it might create awkwardness, is that you'd be bringing up something they don't want talked about. So whether you can raise the issue depends a lot on the personality of the person you raise it with and how you raise it. Everyone else is in the same position that you are, with the same concerns, including your boss. Everyone wants as much information as they can get, about the prospects of the company and themselves. It's natural to ask, and they'll be expecting people to ask, but they also won't enjoy the fact they have to tell everyone who asks that they can't guarantee anything. Make your decision based on that.
ask them if they consider me to still be an asset and needed by the company
Yes, they do. If they didn't they'd have laid you off already. Since they're making layoffs, there is no reason not to have included you if they didn't want you. If you need to hear that directly from your managers, and it will help your feelings about work, then go for it, ask. They will tell you "yes", and you will say, "great, that makes me feel a lot better", and you don't look disloyal.
However there is no assurance that there are more shoes to drop
Correct, and you cannot get that assurance. The company is in trouble, or it would not have already made more than one round of layoffs. The people they laid off, especially in second and subsequent rounds, are people they previously valued and thought they needed.
Aside from more layoffs, the company could go out of business. It could get a big new contract and start hiring again. Realistically the former is more likely than the latter, but you need a job, and no job or employer is secure forever, so you accept some risk.
I've been in this position, although there was only ever one set of layoffs, not multiple rounds. The time from the layoffs until when the company (actually, a sort of follow-on company, the details were complicated) went bust was about 8 years or so. Not everyone plans to be at their current employer for 8 years anyway, and in fact I left 4 or 5 years before the bust. So what the heck. I don't think I played the situation perfectly, but whenever anything ends badly you'll wish you'd ended it sooner. Accept that.
this is impacting my ability to concentrate on my work
You might be able to improve this with some kind of cognitive self-trick like, "I don't have to worry about this. I have a couple of months salary saved. I will work for this company for as long as the job still exists and I still enjoy it, and then I will find another job".
Alternatively, if you simply cannot bear to work for a company that's in trouble, then the job isn't good any more and you should look for another.