Jamie is the opposite- I love the drunken boxer breakdancer thing but the drink mechanic is horseshit, especially since it leaves you so vulnerable AND they don't carry over to the next round like Manon's thing does. I will probably never play this character again.
So the deal with Jamie is that some of his attacks have drink-variations (like Juri's kick charge thing), like his rekka if done with kicks automatically gives him a drink, his sweep has a drink followup too. You are not really supposed to do the down down drink move much at all actually. About the only time you would use it is when someone is legit turtling from full screen away, or when you landed the rush palm OD which has them tumbling away and allows you to get 2 drinks for free. Besides that, the idea for the char is to just fight and get hits and convert them into drink-combos. Also his lvl 2 super instantly fills him with 4 drinks so you can just use that and go to town.
A lot more complicated than something like Ryu where he just dps and shoots fireballs until he gets lvl 3 super, but a lot more fun to execute too. And yeah he is one of the hardest chars in the game.
As for drinks carrying in the next round, they "kinda" do since usually you will have 2 bars of meter in round 2, during which time you get to do this:
I've had numerous fights which started with this exact combo and concluded with a very, very fast perfect, before the lvl 2 super ran out, and a ton more which left me at a huge lead thanks to it.