Many, many mechanics STRONGLY discourage commercial transmission flushes. The reason is that the machines that are used for this operate at a high pressure that can dislodge gunk in your trans. That gunk then may clog passages, resulting in transmission damage. Not good.
Here's a better plan:
Pull the pan, clean or replace the filter, then refill with clean fluid. Now, if you want to get out ALL the fluid, you can disconnect your cooling lines. Make sure you know which is which! One of them will be returning fluid from the radiator to the trans. THIS one goes into a container of new trans fluid. (Hey, make sure it's clean on the outside so you don't contaminate the clean trans fluid!) The other one takes fluid from the trans to the radiator. THIS one goes into an empty container. Preferably, one that's translucent so you can see it getting full, and a bit bigger than the container of clean fluid . Now, have a helper start the engine, and the trans will pull clean fluid in, and pump dirty fluid out. When you see clean fluid coming out, you're done. Shut off the engine and reconnect your lines.
This doesn't have the high pressure problems of a commercial flush, but it will get the old fluid out.
As for the cooling system, you can always cheat. I put in a quarter cup of Amway SA-8 laundry detergent, drive around for a few weeks, then drain, fill with water, run for 10 minutes, then drain a fill with antifreeze. Works great. Recommended to me by an old mechanic 30 years ago, and I'm sold on it. There's nothing in it that will hurt your system, and it does a good job of cleaning things out.
No, I don't sell it. You can find it online.