What to do if I was in a fender bender and the party involved is trying to demand for more than what my insurance covers?

A person is only obligated to pay, assuming he or she is at fault in the accident, for the property damage and personal injury he or she causes. An injured party can sue you for more than your insurance coverage, however--the level of your coverage affects how much reimbursement or payment you get from your insurer, but does not limit what the injured party may claim for; he or she can sue you for any amount of damages which are provably the result of your wrongful actions (e.g. negligent or careless driving).

If his person has in fact suffered $30k or so of medical costs, lost wages, and "pain and suffering," she therefore can sue you for that amount.

First you need to determine what your insurer is doing: are they going to defend you, on the grounds that they are contesting your liabiltiy or fault (though it sounds like they are not) or because they feel she is improperly inflating her claim and is not entitled to that much? Or does the insurer think she validly has at least $15k of damages which are provably your fault, in which case they intend to pay out to their policy limit.

If the former--if they will defend you--you need to coordinate with their attorney. However, if the latter and they believe you owe this woman at least $15k and hence intend to simply pay, then you have three options for the remaining $15k of her presumptive claim (i.e. if she's claiming $30k total and your insurer pays $15k, she can then seek the other $15k from you):

1) You can agree to pay the amount, and work out a payment plan.

2) You can try to negotiage to a settlment, and see if you can resolve the remaining liabilty by paying some smaller amount (e.g. $5k - $10k).

3) You can refuse to pay, see if she does in fact sue you, and if she does, defend against her suit by trying to prove that either you were not at fault or that her injuries are worth less than she is claiming.

And of course, you could initially refuse to pay, then try to settle even as you defending yourself.

The best bet, if the insurer will not be defending you, would be to retain an attorney to represent you--for $15k of potential liability, it will be well worth it. Your lawyer can evaluate the situation and your potential liabilty and advise you as to which course of action is best.