A bit of cold water poured over your head works wonders cooling off. I've got enough hair that will hold it for a bit so it can do its job, but lacking that you can soak a hat or bandana to do the same. It's liquid humid here usually so stuff just doesn't evaporate, so much so that welding gloves and stuff are dried using the radiator exhaust to heat them up enough to dry out. The only way I've found to really do the heat and humidity in this setting is by physically drinking (chugging) as much water as possible (with electrolytes and salt as needed to control cramping), and then dumping water on you to cool off if needed.
Going to sit in a truck with the ac blasting during frequent breaks can do wonders too, 20 min of relaxing, eating and drinking, and soaking up ac every couple hours makes you a new man. I've looked into a cool vest pretty hard, but with me working as a fitter where I'm welding all day i just can't see it lasting or even working in some spots (the ones i always seem to find myself in). I've got a few yeti style water bottles, and I'll fill them up so i got plenty of water with me at all times. It's cool, not cold, if it's too cold you don't drink enough. Some ice cold water is handy to quickly drop temps tho, welding pipeline it was part of the contract that you got ice and bottled water daily for your rig, so i always made sure everyone around me had water. You can always let it warm up a bit if it's too cold.
Welding in the desert where you're at must be hell, but maybe the dry heat part might work to your advantage. Evaporation happens there quite quickly, so your sweat actually does its job. I would likely be in long sleeves to wick water, and would likely be tempted to dump water on my clothes there to evaporate before the sweat, maximizing cooling. Here you would just stay wet all day if you did that, which isn't the end of the world because you'll be drenched anyways from all the sweat that just sits there.
A weld cap or bandanna under the hard hat adds another layer that can evaporate water to cool you off, so that's what i would be headed towards. Every so often stop to drink water, splash a little bit on your head/clothes, and carry on. Crude, but effective, and i haven't found anything better yet myself. I would also be looking into making a redneck swamp cooler fan thing, think 5 gallon bucket, fish tank pump, burlap bag, and a big fan. Definitely a canopy or umbrella too.