This sounds very familiar - I have some issues with social anxiety and am very bad at hiding when I feel uncomfortable, so I've been in a number of situations where my shyness / awkwardness has been noted by other people.
My first suggestion is for next time - If you know or suspect that they think that, I've found it's best to address their misconceptions when it happens, i.e., at the party. While it doesn't happen often, I have been told directly that I look uncomfortable by one of the friend-of-friends... as embarrassing as that is, it did give me a chance to explain and say "oh, no, I'm just shy" and assure them that I really was enjoying talking to them. Otherwise, it's pretty common (in my experience with casual / small house parties in the US) to get a chance to address the group as you're saying your goodbyes for the night - thank the host for inviting you, tell everyone you had a great time meeting them and were glad to have attended, accompanied by a genuine smile :)
However now that the party is over, you'll have to make contact some other way. One thing I've done is to add them on social media, possibly with a friendly message (e.g. "Hi, it's Em! Had a great time talking to you about foobar at the party last week"), and make a post about how I had a great New Year's Eve. I wouldn't message everyone directly to say "Hey, I heard you thought I had a bad time, but I actually had a great time!", since they might feel uncomfortable and pressured to respond. Instead, adding them on social media implies "I'd like to stay in touch", which is a positive and fairly low pressure signal, and the post will make it clear you enjoyed the party.
You also can talk to your friend about it! She's your link to the others at the party, and that's how you found out about this situation, so it's clearly not out of line to bring it up with her. Ask your friend to pass it along that while you're a bit shy around new people, you had a lovely time (would be happy to hang out again, etc., whatever is true for you). This is another thing I've both done and been asked to do by others. I'm not sure it's expected, but I do personally feel some responsibility when introducing a new person to a group to make sure things go well, so I don't think it'd be unusual to ask. And it should be a relatively easy thing for her to do, since she understands the dynamics of interacting with both you and her other friends and will know how best to explain to them.
