Geico can pay only what you bought and in this case was $1000 in medical payment coverage, thus most likely they will pursue the at fault for the $1,000 they will pay on your behalf and anything more, will be your responsibility. Med pays only for any medical bills due to an accident, but does (not) pay lost wages. Now if you lived in a (no fault) state, then you would be required to have PIP, which is personal injury protection that pays for the medical bills and also pays for lost wages up to the LIMITS you bought. Now (if) your state allows your insurance to subrogate for your medical payments on your behalf, then most likely Geico will get a copy to find out who to pursue or collect, but right now there is no hurry for them and they will pursue later.
Sounds like you do not have collision coverage, since it you did, then they would fix your car (minus) your deductible, thus the reason it sounds like they are blowing you off.
Do you have health insurance? If so, then you file for any medical treatment to pay. But if not, then you will have to pay for any medical treatment yourself.
Now depending on where you live, then you maybe eligible for a injury settlement which is (reimbursement) of your medical bills, any lost wages, then (adds) pain and suffering. But that would be paid only after done treating.
But you have a bigger problem that they have (should) make you aware of and that is the LIMITS of the at fault driver who could be liable for (4) car damages (plus) injuries to 4 other drivers (plus) passengers in those other 4 cars.
If the at fault has the minimum limits required by law, then odds are that you (won't) get your car damages and injuries paid in full, since the at fault insurance will have to split out and pay the max and if not enough, then you will be SOL.
Now, if you have UIBI (UNDER -insured bodily injury) then your insurance would take over and cover what the other insurance is (not) enough The key is UNDER, no Uninsured, since even if the other person has insurance they are insured, but have to have Under insured, meaning that the other driver does (not) have enough insurance to pay you or anyone else
If I had to guess, you don't, since if you did, then your adjuster (should) have explained what coverage you bought and explain what (each) covers.
So, when the police report is ready, then call the Geico adjuster again and ask them to call the at fault insurance company (will be listed on the police report with the policy number) to find out how MUCH insurance there is to pay and (if) they will pay in full or not, since 4 other cars. .
If not enough, NO lawyer will be interested, since they will take 33% of any settlement which would be LESS w/out a lawyer. You could sue that person, but no good, since you can get a judgement, but the courts don't collect, thus they could be judgment proof, and never see a dime of it.
I know that you are PO, but you need to understand, that you need to buy enough insurance for YOU, since so many times the other person has no insurance or not enough, thus you may see that you could be SOL.
As a retired auto adjuster seen this too many times. I have 1/2 million in liability plus a unbrellla for 1 million. I have $10,000 med pay (plus) Medicare. I have (both) uninsured and under-insured bodily injury for 1/2 million and uninsured and under insured property damages for my car and collision and comprehensive.
The uninsured and under insured cost very little to add to the policy, but I will never ever reject or take it off.