Maybe I am not sure what you are asking here.
If you are asking, “Can I drive safely and call in a report of a drunk driver at the same time?” I would say almost certainly yes. Hell, until it became illegal to drive and use a cell phone most of us probably did and most of us survived the experience with no trouble.
Certainly it is a good thing to call and try and get the drunk off the road. Then again this article suggests using a cell phone while driving impairs you as much as being drunk. To be fair though that is only when a person is at the border of the legal alcohol limit which, obviously, is definitionally not impaired (or impaired to any significant extent). If you were too impaired at the limit then the limit would be lower. Personally I think you are fine calling for this purpose.
My point is that when you call the cops to report the crime (remember 911 calls are recorded) when you say you are following a drunk driver you have just admitted to the police that you are committing a crime too. Maybe not a terrible crime but a crime nonetheless.
Yes it would be shitty for the police to issue you a citation but they might. If you do happen to get in an accident while you are doing this (unlikely but possible) you are in bigger trouble (your insurance will probably deny money).
If you think it important enough to call it in anyway and take the (probably small) risk of the above knock yourself out.
Or you can collect the relevant details, pull over and call it in. You are not the cops. It is not your place to chase suspects.