No, 'fraid grocery shopping doesn't count as time away (though it's nice to get time alone to shop, too).
You need time AWAY, to not be Mommy at all. Is there another mom with whom you could trade afternoons watching the kids? Is there a Mom's day out at a local church?
And as long as it's not so hot out that there's a health warning, kids can play outside. Humans haven't always used air conditioning, after all! Maybe send the kids out in the morning instead of the afternoon, and use plenty of bug spray and sunscreen, but trust me, it can be done! My kids just came inside, and it's 90+ degrees out. They played in the shade and they didn't do much running around. But they went out and had fun without me, just as they do pretty much every day, rain or shine.
As for your DD...What about trying a little reverse psychology on her? "No, you can't use this playdough. It's only for older kids...Well, okay, you can use it, but you'd have to do it alone while I work with your brother. I don't know if that's okay with me...I really ought to be watching you to be sure you don't mess it up...Alright, I'll trust you with it, but I'm not sure about this..."
Or is she too young for that sort of evil-mommy manipulation? It works well on my 4yo.
Some of this is, I think, just the attitude: look for ways to make it work, rather than for reasons why it won't. You may have problems to solve (mosquitos, heat, no $ for sitters, etc.), but there's a solution that will work for you if you look hard enough for it. And for me, it's been worth the effort! Maybe you're doing better with it all than I was, but boy, I needed to find ways to make it work.
No, you're definitely not the only one who has struggled!
It's so much better for me this year than it was for me last year though. That's the good news!
