First, brace yourself for a 3 year battle. Second, there's a few options you can pursue, and I'm answering this under the presumption that the paternity issue has already been settled.
First petition for temporary visitation while the matter is being disputed. Unless there's a good reason not too, a visitation order will almost certainly be entered in your favor.
Second, if she continues to withhold, file a rule to show cause (Generally Indirect Civil Contempt, but criminal is also an option), and visitation abuse. If you prevail on the show cause, you get your attorney fees back for both (Which is nice). Visitation abuse is an expedited process, the rule to show cause will take longer. You're allowed to do both.
Either way, get a lawyer, it's going to be a dogfight from here on out.
Unfortunately, your situation is perfect proof why oral agreements regarding children never work. Where are you in the process of establishing custody because you didn't say. Do you have a current visitation order, and is she violating it? If so, the other attorney is right that a petition for visitation abuse will help you.
As for the drinking, ask the court to order neither of you consume alcohol 6 hours prior to of having the child in your possession, nor anytime that you have the child in your possession. No court wants to have a parent drinking alcohol in the presence of a child.
Does this child go to daycare? Talk to the providers, and find out how the child acts, or if the child says anything about Mommy acting "funny" when she drinks "smelly stuff" or "big people drinks," as that is how children usually identify alcohol.
If your child is crying on the phone, the court needs to know this as well,
Work closely with your attorney. All this could really help your case if you're willing to make the effort to show the court all that you've mentioned above.