I don't see any reason to delay training...assuming the legalities have been resolved (sounds like they have been) and you can get a class 1 medical, etc. The biggest thing will be just putting time between the situation and when you get your major airline interview.

I think what some of the other posters are getting at is that it sounds like you're rationalizing. Obviously I don't know you and maybe you did get a raw deal but doesn't really matter, you just have to own it and move on. Ideally, when you interview for a legacy 10 years from now (or however long) you can say something like this, "Just after college, I was arrested for drinking and driving. It made me realize how irresponsible I was and how much worse this could've been (etc.). I learned X, Y, Z and changed my behavior (as you described above). It's been over a decade since that time and I haven't so much as gotten a speeding ticket." If you can say some like this then I'd say it's a non-issue. Now you can probably get a few traffic tickets or whatever but if you get another alcohol related arrest I'd say you're probably done with top tier airlines.

FWIW, I know two people flying the airlines with previous DUIs in their younger years...one has been at a legacy for awhile now. Good luck!