I am being sued for some unpaid dental bills. My question is that if I pay them now that a lawsuit has been filed, is there any way to avoid a judgement being listed on my credit?

If you pay them now plus all of the extra fees not added ( most likely attorney fees, court filing fees) you can get the case dismissed. But most likely they will want them paid in full.

You could also contact the representative for the dentist and enter into a written settlement agreement that could include, among other things, that the claim be withdrawn and removed from your credit record upon final payment of the settlement amount.