Will an old criminal conviction keep me from adopting a child?

I was convicted of felony drug possession and distribution when I was 19 years old. I served my sentence and had a perfect record on probation. I’m now 34 years old, married, and have had no other trouble with the law whatsoever. My wife and I want to adopt a child but the worker at the Las Vegas adoption agency we met with said my felony conviction might bar me from adopting in Nevada. Is that true? Is there any way I can get the conviction off my record since it’s been so long and I have had no other convictions or legal problem at all? Could the same lawyer help me with both the adoption and clearing my record?