I have some 45 years experience of folk clubs, singarounds and sessions, and nearly as much of folk festivals.
Festival singarounds are always very crowded, but eventually everyone who expresses the wish to sing DOES get the chance.
There are a couple of useful tips tho'
1. Arrive early. If possible, a little before the start of a session. The organisers do take note of people who are there before it gets crowded, and that almost guarantees an early chance to perform.
2. Choose your BEST songs. If you impress with your first, they will want to hear you again, and that twig will come back to you at some point.
3. Join in choruses. If you know the song that someone else is singing, and can add to the mix, it is almost like performing yourself, and it shortens the wait till your next opportunity.
Regarding your last comment, I have never met a group of folkies who wouldn't welcome a lone individual into their circle, if they become aware that you are alone. Folkies are, by the very nature of what they do, gregarious. Take any opportunities to talk to the people around you, and I'm sure that you can establish a presence within the group.
I wish you all the best for your experience of Sidmouth this year. Dive in and enjoy Don T.