I agree with everyone here - my exA just quit the other day. I thought hard about your proposal to agree to random testing with your husband, and then I put that shoe on my foot. It's a tempting offer and appeals to my control-freak side but, in the long run, actions really do speak louder than words. And, truthfully, quitting drinking does not solve the problem (although it is a start). Recovery is what will provide the soil for confidence and faith and trust to grow - I firmly believe that.
Congratulations on taking the first step!