If you're using the (e)at command to drink blood off weapons or the ground and such, that doesn't work (and that seems likely, since you end up full). What you have to do is approach a sleeping or unconscious NPC (yes, choking them unconcious, or various other violent means will work, if you don't kill them first) and then use the (e)at command, and an option will be on the list to "Drink Soandso's blood" or something to that effect. I've never gotten a message of being "too full" from it either. It is possible to drink until you drain them dry (takes a very long time in most cases, unless they've already lost a lot of blood from injury).
Just be warned: If you are spotted drinking blood from your victim by another person, even one of your own companions, everyone in the immediate area instantly becomes hostile, including your companions, and you'll be marked as an enemy of the civ.
As for skill increases, your physical attributes will be locked when you become a vampire. Strength, Agility, and so on will no longer increase. Mental attributes should still slowly grow. Skills like weapons and Fighter should still increase, but as far as I know, you gain nothing if you're hitting an unconscious foe. I usually don't let them lie unconscious for long, and finish them off with a good curb stomp anyway.