If you had personal injury protection (p.i.p.) coverage then your insurance should cover the medical bills.
The driver of the other vehicle will only be legally liable if he was negligent. His liability would be proportional to his percentage of negligence. The owner of the other car, assuming its not the same person, may also be liable.
The ticket is totally irrelevant to establish liability. The fact that you were ticketed shouldn't even get before a jury. Most laymen, especially adjusters, do not know this.
You should schedule a formal consultation with a lawyer. A good means of obtaining an attorney referral is by contacting your State's bar association or your local county bar association. They probably have a web site and charge a nominal fee for the referral. Most attorneys either provide a free initial consultation, or charge a nominal fee.
Good Luck, Elliot Jay Goldstein (Attorney with offices in Tampa and St. Petersburg - Feel free to contact me directly by sending an e-mail via clicking on my profile tab)
The above information is provided as a courtesy, without any consideration and without knowledge of the specific facts. Do not rely on this information. Seek a formal consultation with an attorney.