It is not really a common practise, but companies do tend to pay full-time salary to people who they feel are working well; so that they don't lose out on them. (Sort of retention hack)
Even my company does it. So, I would answer your questions one by one:
Is there something I should be asking the recruiter to find out if
this is a real deal, or if there's something awry?
Yes, you should ask the recruiter about that. I am pretty positive that it is for retention, but you might want to confirm that from the recruiter or your manager(if you have any).
I suppose my biggest concern is that if they can't find contract work
for me, will they really continue to pay me or will I get laid off
NO They wouldn't continue to pay you if they can't find a contract for you for a long time. The full-time pay would only till they find contracts for you to work on with short intervals between them. Cause it wouldn't cost the company much, and they are retaining a nice candidate. So, a win-win for both. But, for a long-time? No. Even though you are a spectacular employee, they can't and wouldn't afford to give you full-time salary for nothing.
Would it be appropriate to try to negotiate a severance contract if
going that route?
Yes, you would definitely want to. As I mentioned above, in case of a long gap, you might be laid off. So, do negotiate a severance contract forehand.