I have a bunch of questions I want to ask you and then I'd like to offer advice if I can...
When you say drooling, is she opening her mouth and dropping a load of water? You said you just opened your pasture, is there a lot of clover in it?
Have you had the vet out since you bought her? I would have her teeth checked, that's first and foremost. Clover will cause a horse to drool.
Sometimes just giving them that fresh green pasture in the spring will cause them to start feeling their oats and they will act out.
What breed is she?
You might want to look at her withers and see if her saddle is rubbing her at all. Some of it could be pain caused by a poorly fit saddle. What type of bit are you using?
At two I would probably say that more than likely it's an age related issue. Can you tell me some more about this trainer? If she almost got you off, let me ask what you did at that point? Did you dismount and put her away, thus ending on a bad note or did you continue to work her until you had a good note to end on? With a baby you could potentially teach her that acting out will gain her a reward if you quit on a bad note.
One other thought, that 12 year old horse may actually be better for you, but I'm curious about this 2 year old, it isn't exactly the best horse for a green family to start with but it might be a nice enough quality that you could put 30-90 days professional training on it, sell it and buy a really nice horse, OR, after 30-90 days training it might be a keeper.
You might want to look into lessons for yourself as well...
I'm curious to see your responses!
"Be a best friend, tell the truth, and overuse I love you
Go to work, do your best, don't outsmart your common sense
Never let your prayin knees get lazy
And love like crazy"