Old enough that I've been there done that like some others on here. First time I lived in Texas was in the mid-70s. We had a fan or two, but A/C was not a part of our life then. I rode a motorcycle all over the place and wandered around on foot. I was also young and fit. Being out on the bike, horse, and on foot, probably acclimated me the most. Both back on the farm in southern Ohio and there in south Texas I and others loaded hay the old fashioned way in the summer heat. Dragging heavy rectangular bales, slinging them up on to the wagon, then going to the barn to load them into hot hay lofts.
When I moved back to Southern Ohio, I worked in a factory and share an cheap second story apartment with a guy who worked there too. We went through the winter with no heat except for maybe three days the stove worked. No cook stove either. Adapt and function. Usually ran out of food by Wed night or mid-day Thurs. First stop after cashing our paychecks on Friday was the DQ. Small town, not a lot there. And it was walk to get where you were going.
When I moved back to Texas in 96 it was brutal. I'd gotten used to and liked the weather in interior Alaska and the Willamette Valley of Oregon. At first we didn't have A/C at all. I had come back to marry my HS sweetheart. She had three kids, one with severe autism, and only her to support them. I got a job as a photographer for the local paper and didn't make much either. The kids slept in the one bedroom house and she in I in a shed she and some friends had built. Yeah, it was freaking hot.
I did find that once I got into motorcycles again and went to that as my only transportation for a few years that I acclimated better and the heat was less troubling to me that when spending most of my time in a car with A/C. Reminds me. I need to get the bike on the road more before it gets fully hot full time. It's already broke 100F here already in early May.
Back before I went active Army I spent a few years in the reserves. My last "summer camp" was at North Fort Hood. We got to go out in late June to play in the field for a few hours at MOPP4. If I ever end up in an environment that requires being in MOPP4 as a way of life. Just shoot me.
I spent a year in the Sinai as permanent party at North Camp. Yes, I loved my window A/C unit in my room. However, I saw plenty of people, locals at El Arish and bedouins, who lived in that environment fulltime and functioned by working with the heat, not against it. The soldiers from the Fijian contingent would be out doing PT, running, and doing log drills in boots, fatigue pants, T-shirt and web gear. Those boys were FIT.
In other words, people can and have functioned in extreme environments for a lot longer than they haven't. As others have mentioned, stay hydrated, stay out of the sun as much as you can. Find shaded places with a breeze if you can. If it doesn't need done, don't do it. Also watch out if you are in places of high humidity. That can zap you. Or if you are used to a high humidity area and know when your body is telling you that you are overheating and go to a low humidity location be very aware. You won't get the same early warning signs as usual. I went from Fort Bragg, NC to Nordstrom AFB for Gallant Eagle 86. I was used to the humidity at Bragg and while the low humidity at San Bernadino was great, it was different. We were putting up GP mediums and doing other stuff and I got hit with the heat. Didn't see it coming because I wasn't getting the usual indicators.
Funny thing is, my heat injuries have happened at lower ambient temperatures and even on some overcast days. I've been zapped in the upper 80s and lower 90s with overcast, but high humidity. It's never been when it was 100 plus out. Maybe because I was more cautious and aware, or perhaps other factors. Just an observation. Just like hypothermia, you don't have to be at sub zero temps to get there.
If you have water available to spare wet some clothing or clothes and let them evaporate to cool you and others. Think in terms of what holds coolness and what acts as a cooling process. Remember, most caves have a constant year round temperature in the 50-60s. There are a few that have and require higher heat and humidity. The Caverns at Sonora being one. So think in terms of in the ground as cooling. No is the time to try out ideas and see how they work. Also, look at why you would need to relocate so far away.
Unfortunately, we have become dependent on air conditioning and central heating. Our houses are generally built to take full advantage of that and not for good air circulation to aid in cooling without. Really think about would it truly be life threatening, or just uncomfortable. Can you alter your routines and adjust, then deal with being uncomfortable, but able to stick it out? If you don't need to move, don't. Around the house and especially moving from the house. Trying to bug out in massive heat without a lot of careful consideration and planning can, pardon the pun, put you out of the frying pan and into the fire.