If I have a son and the mom wants me to sign over my rights so that her husband so he can adopt him, will this eliminate my financial responsibilities?

Typically, terminating your rights relieves you of any obligation to your child from the date of termination forward. However, relinquishing parental rights isn't something that the courts take lightly. Although not always stated, the general standard used by the courts is for "good cause". The fact is that relinquishing rights for adoptive purposes is seen in a better light, especially sine it means that there will be someone else to take over your child's support.  

Note: It is doubtful that any jurisdiction would relieve you of past support, if any.