I think you're right, that the problem is lack of confidence. Taking a friend along, like Sleepy Hollow suggested, might help with that. Also remember that the other people in the class are probably not even thinking about how you're doing with the class work, because they're also trying to keep up. It's not unusual to feel a little frazzled when you're trying to learn something in a group, because everyone learns in a different way and at a different pace. Don't be so hard on yourself!
I sometimes feel like I've fallen behind in class, but then I look around and can usually spot at least one other person with that "deer in the headlights" look (sometimes it's the whole class). That means it's time to get the instructor's attention to help us out. I don't expect to be the one who finishes first, or the one whose project looks nicest. As for equipment, I usually bring a cheap older (not vintage) Singer, rather than lug my better machine to class. And unless someone else has an unusual vintage machine, I don't notice what machine or supplies they are using. If the shop owners are making you feel like your machine doesn't match up, then shame on them!