If I sign 2 type A Free Agents and lose 1 type A Free Agent, what draft picks am I with? Do I lose my first rounder and get a comp pick or does my type A and the one I signed cancel out?

You will lose your first round pick if below #16, and your 2nd round pick. You will gain a 1st or 2nd round pick, along with a compensation pick, from the team that signed your type A. The pick depends on where the team is listed in the draft.

to elaborate on yoker's post...

if you're first round pick is better than 16th then you would lose your 2nd and 3rd round pick...if it's worse then you would lose your first and 2nd rounders...

you definitely get back a compensation pick between the 1st and 2nd rounds and depending on where the other team drafts you could receive a pick in the latter half of the 1st round, or the first half of the 2nd round...or if they signed multiple type a's (like you did) you could get a 3rd rounder (or even worse)