It’s hard to give an opinion without knowing the facts. The personal injury lawyers you spoke to may think the case is too small to take OR they may think that you don’t even have a personal injury case (because you can’t prove that somebody else is at fault for your slip and fall).
Assuming you do have a personal injury case, I would tell the claims adjuster that you want to gather your medical bills and reports and that you will submit them to him/her. You’ll actually have to get records of your treatment from the doctor and/or hospital. When you submit them to the claims adjuster, let them know you want your medical bills to be paid PLUS something for pain and suffering and/or a possible scar on your hand. You WILL have to reimburse Medicare/Medicaid. One other tip: do not agree to give a written or recorded statement to the claims adjuster.