Think about mixer dances. If there are wallflowers, call Dashing White Sargeant, Twelve Meet, and my favourite Horses Brawl.
Specifically state that gender is unimportant and encourage the timid to just join a set, or get couples to entice a third.

With Horses Brawl you can do it with 7 to 10 (depending on skills) and partners are irrelevant. And it is a simple dance in essence.

One dance I went to the band were, hmmm, probably a reasonable Concert band doing Irish. But before we discovered their historical experience at English Ceilidh (hmmmm) the caller started with total novices and us, by saying:

"I want you all to say hello to each person sitting (they were sitting) next to you, now hold the hand of the person on your left, and now on the right, keep holding hands. Now stand up and follow the line, don't let go, don't let go"
he had held the hand of the first person and pulled the line on to the floor and the band started a simple tune as we snaked around the floor looping under a pair of arms then another.
He described this as a Farandole. Well an Anglicised version thereof, but it got everyone on the floor for the first dance! Ice broken! I was impressed. Unlike the bands attempts at matching the music to the dance, or our feet.
Keep that one in your pocket for difficult starts.