My first thought is you got the lemon in trainers in your training class if that's what they told you to do and it was their only idea.
I agree with the socialization, most any dog benefits from socialization and habituation.
I strongly, strongly, strongly urge you to find another training class, with a different trainer. Piper is just a puppy so it's not too late to teach her proper manners and key to that is YOU must learn how to make yourself more attractive to her than those other dogs and people. That's a very tall order.
Some ideas:
How much off leash exercise does Piper get? Terriers have a lot, a LOT, of energy. Leash walking alone is not sufficient for most dogs. A good run off leash DAILY will help take some of the starch out of her and make her more likely to listen to you.
For distraction like that I found Leslie Mc Devitt's book, "Control Unleashed" worked wonders for us. BUT, my dog doesn't bark at people. I think you'd need a good trainer working with you to make sure you don't inadvertently reward the barking.
I will see if I can find it, there are training sessions for exactly this kind of behaviour somewhere on the internet. One is at Sue Ailsby's website and maybe I can link it. There's another one where you need an extremely patient friend, a not well used place, and you simply have the person stop approaching you the nano second the dog begins to act up. It can take hours upon hours.
Here it is, Song and the Sheep from Sue Ailsby's dog training site. You will see this also took hours but it is similar to your problem:
http://www.sue-eh.ca/page24/page38/
Good luck. I wish my neighbour with the exact same problem would work on her big Goldendoodle. She is also embarassed but she doesn't seem to put any work into helping her dog at all. Rather be embarassed I guess.