malpractice suit agaist my lawyer without legal representation

YOu are not required to have an attorney, but you should discuss you claim with an attorney, at minimum to properly evaluate the claim.  Since your claim is against another attorney, some of those you call may be reluctant to assist you, but you should be able to find someone with whom to discuss your claim.  

You can file any kind of lawsuit on your own, but I wouldn't advise doing this one that way.  In the first place, verbal agreements aren't reliable.  More importantly, malpractice is a fairly specialized area of the law, and while your case is a relatively simple one, as far as liability, you'd no more want to try this case yourself than you would have wanted to try the auto accident case, particularly on damages.

The suit will be defended by the insurance company's lawyers;  they've done this before and they do it for a living, and they will spot any mistake you might make and take full advantage of it.  And to prove your damages, you will have to have a "trial within a trial," to prove what you probably would have won, if your accident case had been properly filed and pursued, complete with (possibly) medical expert testimony about your injuries, etc.