I was involved in a small car accident with another driver. Apparently, my insurance coverage lapsed because of a late payment. What can I do?

As you suggested, the issue of who is responsible for the auto accident is a question of "he said" this and  "she said" that.  It's a question of fact -- and each driver likely has a different perception as to who was at fault, and what share of the fault each driver bears, and whether the accident could have been avoided only if the other driver had done such and such.

If there were any witnesses, each of them also have their own views as to what happened. Further, where there is an accident, may states require that an accident report form be filled wit the DMV within a fixed number of days if there is any personal injury or if property damage exceeds a certain dollar threshold.  Other states require reports be filed with local police officials.

If you were not at fault, you should have no liability as a matter of law. Depending on your state's laws, if there was at least partial fault on both sides, the person who is more at fault (or his or her insurance company) generally is liable for a proportionate share of the damages. (In some states ANY fault on the part of a driver bars any recovery from a third party.)

The difficulty is as perceptions of fault can vary over time, the mere fact that you don't think you were at fault does not mean that you were not at fault. Even if the other driver said at the accident s/he was fully at fault, there may be nothing stopping him or her from later claiming s/he never said that and suing you. Even if there was a police report of the accident the conclusions of the police officer are not determinative either.

The major problem with any auto accident like this is that while it may look now as if the only damage is property damage to the auto -- and thus the amount of the damages seems rather easily quantifiable (what would it cost to repair the car) -- there always is the potential there was some personal injury not recognized at first, and for a major claim to be asserted.

It is not unknown for driver 1 to pay the costs to repair the car of driver 2 out of pocket, only to find it followed by a claim or lawsuit for huge personal injury damages down the road unless an effective general release is obtained from the other driver and every passenger in both cars. 

If you had liability insurance, your carrier would defend you and pay all claims (up to the limit of your coverage), and deal with the other driver and his or her collision insurer. (If the other driver has collision coverage, s/he most likely would have his/her carrier deal with it and initially bear the deductible. His or her insurance company would then have a subrogation claim against you and come after your insurance company for what it paid out plus the deductible, keep what it paid out to repair his car, and then make their driver whole by reimbursing him for the deductible it collected from your company.

Yes, it is time to consult a lawyer, and our affiliate www.AttorneyPages.com is a great place to find a capable lawyer. The lawyer can help defend you, advise you as to whether to make an offer or not, and protect you by making sure you get a full release if you do make any payment. The lawyer may also help you make sure whether your insurance policy actually did lapse or whether it might (to your surprise) be in force, perhaps because the insurance company did not send you proper notice of the lapse.