What can we do to our insurance company for failing to send our policy limit to the other party lawyer?

Its too soon to really answer the question of whether or not they have opened you up to liability. It is not uncommon for attorneys to file suit because the insurance company was not making good offers before. You would only be exposed to liability beyond your insurance coverage of the case went to trial and the jury rendered an excess verdict. The fact is, however, 99% of cases are settled before trial. Even if it went to trial and an excess verdict was rendered, it MAY actually be your insurance company who is liable for this, if they could and should have settled the case on your behalf (these are called "bad faith" cases). If the accident/injuries are severe, and you think that the damages may exceed your coverage limits, you may want to retain the services of a personal injury attorney, or one who practices the defense of personal injury cases, to represent you by walking you through the process and making sure your interests are protected. You would likely have to retain them on an hourly basis.