What would happen if my lawyer could no longer work on my case?

If your lawyer cannot work on your case they are obligated to tell you so and should present you with options.

We have no idea. Your lawyer will tell you.

A different attorney in the firm would take the case. If there was not another attorney in the firm who could take it, then you would be called and told to find another lawyer. If "cutting your losses" is how you view hiring a lawyer to obtain what you are entitled to, then why did you bother in the first place?

If they were unable to keep working on your case, say, because they had health issues, to be professional they should refer you to a colleague who could take over. If you had paid them a retainer that was more than they had already worked, they should give you a complete accounting and any remainder so you can pay the referral attorney.

Presumably, the referral attorney would likely be another at the same firm, but if the others were not able to take your case, they might refer you to someone else. Without more details, we can't really tell you, but that's something you need to discuss with your lawyer. They have a professional responsibility not to just dump you if they are unable to finish the case.

They would probably keep it in house. But they would have some reccomendation.

Who knows. You aren't giving us any reason WHY your lawyer can no longer work on your case. Yeah they could offer to turn it over to another lawyer in their firm. Although if they are DROPPING you as a client then that's pretty unlikely.