Yeah, sure, you can absolutely move if you aren't happy where you are. It doesn't sound like anything is forcing you to stay there.
Otherwise, be creative.
I'm friends with A LOT of overworked parents who have no time. I make sure that they know that I'm available for last minute things, that their house can be a mess and I don't care, that I'll come over and cook for them on a weeknight if that's what would help them, and I've even just accompanied some of them to the grocery store to help them wrangle their kids while we shop together. I'll often volunteer to do a meal plan for them of easy, low cost meals.
My sister is doing an intensive degree while working full time, so she's crazy busy. I have her over while I batch cook, she chops a few things and does some dishes, but mostly just vents about her work and asks advice on her projects, then we go for a quick swim in the pool, and I send her home with lunches for the week.
Low cost, easy, healthy meals are my passion, which is why my suggestions revolve around food, but you can extrapolate that to whatever works for you. If you want to be involved in the lives of very busy people, then find out the pockets of their lives where you can fit. People usually desperately want friends in their busy lives, but don't have the energy for traditional adult socializing with all of it's pressures and bullshit.
You can also add more seniors to your roster of friends. My favourite daytime hang out buddy is 67 and lives a few blocks away. She's got way more energy than I do though, so I often have to get her to scale back whatever plans she wants to get up to.
DH and I are also friends with some really young people, the kind whose socializing has virtually no expectations, is often last minute, and who rarely say no to "hey, want to go to a free krav maga class?" on a week night.
As someone already mentioned, volunteering is great for kind of forcing friendships, lol. I've been on the same volunteer exec team for 3 years, and despite not overly liking each other at the beginning, we're now all pretty *ride or die* for each other. It's like a family, we drive each other nuts, but the loyalty is very real. I also see them all the god damn time. They're all overworked and have no time, and yet we spend a lot of quality time together and have a lot of fun.
No matter what, you are going to have to swim against the current somehow if you don't want to live the same life as everyone else in your cohort. That may involve moving, it may not. Figure out what life you want to be living, and make changes accordingly.
However, unless you move into a frat house or a retirement community, I think it's going to require creative and active solutions for you to get around the "everyone is too busy" challenge no matter where you live.