Hi S H,
I definitely feel the way you do and I only have one child. Before I had her I worked full time, wore makeup and did my hair every day, and did all kinds of social things. But now I stay home with my daughter, who due to my health will likely be our only child, and spend so much energy taking care of her that I never have the energy for makeup or a cute hairstyle every day.
I usually feel honored with an invitation, so I used to always say yes, even if that meant multiple commitments in one day. But when the day approached I would get that overwhelmed feeling and just want to stay home. I've learned this about myself and now only commit to less things, trying to give myself a day or two of "home" time between playdates, etc. For example, I've stopped committing to anything for the 2 days before long trips (we periodically visit family on the East Coast), because I'd get so stressed out I'd wind up in tears. But when an event is important (to me or to someone I care about) I do just have to acknowledge my feelings and then make myself go anyway. As someone else said, I do wind up having fun and being glad I pushed myself. I think the key is less things and making sure to go to some things. I will admit that I've said no so many times to some old friends that they don't invite me anymore. And that's not a good outcome. So just give yourself some balance. And I don't think you have to go out at night -- I almost never do. I agree, it's hard to have energy for that at all. Maybe try to get together with those friends on a weekend instead when you might have more help with the kids and/or their schedules can be a little more relaxed.
I hope this helps a little. You're definitely not alone!