I'm not working now, so my question is what happens if I sign and send
it in, and get a better offer somewhere else. Am I liable at all to
It depends on the local employment laws, and depends on the wording of the contract you signed.
In most locales, and in most circumstances, you are not liable at all.
You certainly should continue to look for a job while your offer is still contingent. And should you find a job that fits your needs, you can accept the new one. As soon as the new one becomes official (not contingent again), immediately notify company 1 that you have accepted another offer. Make sure you thank them for their consideration, and indicate that unfortunately, you couldn't wait for them to finish their process.
Company 1 may not be happy, but you simply can't wait forever and hope that everything works out.