Making new mom friends - help!

The town I live in is absolutely crawling with new mamas and families with kids. There is so much to do here with a family - free outdoor concerts, family friendly breweries, and tons of "mommy and me" type classes. My question is, how do I turn all these mama strangers into new mama friends?

I've done the classes, and the other moms are always pleasant enough, but it's not like there's an obvious opportunity to really get to know one another - and I'm super scared of coming across as needy or creepily friendly! My sister always said that when she had a baby it was the easiest time in her life to make friends, but I haven't found that to be the case at all.

How do y'all handle this? Would you be totally weirded out if some rando mom approached you and tried to IRL friend you?? My husband, our seven month old, and I have been in our town for almost a year, and we are ready to find our people!