I would never shave my long haired cat unless it was necessary. I have trimmed her fur a little, and use a de-shedding tool to cut down on her fur. The Shed Monster is an economical version of the $30-$40 Furminator, and works well.
I cool my cat down two ways. One is with a wet towel, wrung out. I lay it over her, and wipe her down, including her tummy. She seems to appreciate that, more than spraying water directly on her.
The other thing I’ve found that gives her a cool spot to lay on, is gel pad that stays cool without any refrigeration. I bought one for myself many years ago, and recently saw them advertised for pets. The prices are all over the map, but I was able to buy a large one for $20 on EBay. They come in a range of sizes. I saw the same one I got for $20 selling for $70, so it pays to search for best price. There are several brands to choose from, but basically the same material. The one I got is the “cool mat” from Comfort Ease.