The answer is that you never speak with an insurance company directly. A local lawyer can take the case on a contingency fee basis and get you the money you deserve, not a nuisance settlement.
The short answer is yes, but the reality is probably not. Insurance companies typically like to settle in a lump sum. Your are risking leaving significant money on the table by doing it yourself. You should get a lawyer!
Keep trying to find a lawyer. I can't recommend trying to settle it yourself. Normally, cases don't settle until you have a complete idea of the extent of the injuries - after the healing process has been completed. Then you have a better chance of obtaining a full financial recovery, including the lost earnings, the compensation for your pain, and the medical bills, etc.
[In accordance with the community guidelines of Avvo, this is not "legal advice", nor does it form an attorney-client relationship.]
This is not the type of claim which you should attempt to negotiate on your own. Generally an insurance carrier will not make the type of agreement you are talking about. You should keep in mind that the adverse insurance carrier is neither your friend nor your good neighbor. The sole goal of any insurance carrier as to pay nothing or as little as possible on your claim. Whoever told you that you could anticipate 6 months to recovery after rotator cuff surgery is being very optimistic.
I suggest you redouble your efforts to locate a personal injury attorney. Based on the facts you have told us, you should be able to interest an attorney in undertaking your representation. Don't give any further statement or information to the adverse insurance carrier without speaking with an attorney. If you have given them access to your medical records by way of a release for your records, I suggest you formally revoke the medical release. Your attorney should control the flow of medical information to the adverse carrier.
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Mr. Lundeen is licensed to practice law in Florida and Vermont. The response herein is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, could significantly change the reply and make it unsuitable. Mr. Lundeen strongly advises the questioner to confer with an attorney in your state in order to ensure proper advice is received.