Wow, I feel so bad for you. We live on an island of 5,000 people and actually have the opposite problem Too many friends, too many playdates, no time for schoolwork! It's odd that it can be so different depending on where you're located.
I didn't know anyone when we moved here 10 years ago, but once I had DD I connected with other people who were just starting their families. Our kids have all played together since they were little and we are still very close with the families whose children attend school and the ones who homeschool. Interestingly, although we have a homeschool group on the island, we only really socialize with 2 other homeschooling families and they're ones that we've known for years. Perhaps the difficulty comes with trying to join in with already well developed groups. I know that, unfortunately, we wouldn't really have time to set up more playdates if a new family arrived. DD already has 2 full day playdates each week plus a full day in town for art and music. She also goes to Brownies, gymnastics, ceramics and a 1/2 day quasi-school day with her 2 other homeschooled friends. Plus there are quite a few kids on our street so as soon as school's out the kids are running around the neighborhood playing.
I'm afraid I'm not being very helpful but I do understand. I moved to England in my late 20s and it took forever to make friends. I was only just starting to feel like I had a social circle when I left after almost 2 years. Moving really sucks.