My beagle has been acting a little unusual lately. The back of her lips on both sides appear to be drooping and she is drooling. She never had a problem drooling until a couple of days ago. She is constantly licking the sides of her mouth and opening and closing her mouth. It also appears that she has lost a lot of coordination because when I throw her a treat she will sometimes fall flat on her side or trip over herself.

The increased salivation could be due to periodontal disease, check her teeth and see if there is an excessive amount of tarter build-up, foul odor, or if her mouth seems painful.
I'm only guessing, that may not be the case, but it is something that you can quickly check at home, if she will let you. Periodontal disease may also make her not want to eat if her mouth is painful.
As far as the loss of balance and falling over, that could be many things, from an ear infection to something neurological.
Best to book her in to see the vet.
How old is she and when did this start?