You will have to contact the county clerk for the county in which the hospital is located and check to see if the hospital has filed a lien. Some counties have this information online, some will give it to you over the phone, some you may have to go to the county clerk's office and check.
Lots of different factors affect compensation for a case, the main one being how bad someone was hurt in the accident, how much were their medical expenses, have the injuries healed or are they ongoing, whether a person has health insurance, time lost from work, and the list goes on. These all factor in to the amount of damages. The second question for compensation is the level of insurance available. If someone has a lot of damages, but the responsible driver did not have much insurance or...
Short answer - you can sue both. You need to check if the drunk driver had his own insurance, that might provide some coverage. The owner can be liable for "negligent entrustment" or letting an unlicensed driver or someone he knew was a risky driver operate his vehicle. Hopefully one or both of them have some assets you can recover.
Sounds like you will need an attorney for this one. The attorney will have the resources to find out the owner of the car. There is also a legal issue as to whether the driver was an excluded driver or a non-listed driver, and that can make a difference as to your claim.
Also, I am guessing that you do not have uninsured motorist coverage. If you do, you can make a claim on your own insurance for your damages. Your insurance company pays you, then your insurance company will try to...
Keep calling and pushing the insurance company. You should also consult an attorney; maybe they can help you with this for a small fee. Also, be sure not to overlook your claims for loss of use (rental rate for for all the days you didn't have a rental car) and your claims for diminished value (the reduction in your car's value due to it being in an accident).
Here in Texas, if he is able to get a judgment against you, you could have your driver's license suspended by DPS. Here is a link describing various license suspensions. Click on "Liability Judgment" So, even if he can't collect on the court judgment directly from you since it seems you have no non-exempt assets, you can still have your license suspended until you meet the requirements for reinstatement.
If you were a guest in someone else's house at the time, their homeowner's insurance probably has some coverage for medical payments for people hurt in the house, regardless of whether or not the owner was negligent. You should try to get the name of the owner's insurance policy and file a claim.