I think your radiator shat itself. The transmission does have a cooler around there too so maybe that could be the cause like you said. Also, your trans may be iffy feeling because it's low on fluid. I personally didn't secure a trans cooler line properly after a repair and when I saw smoke and all the fluid everywhere, I thought I blew up my transmission. It shifted much better after I changed the filter out and resealed the trans pan (From what I could tell. I think mine needs some timing work done so I haven't driven it since August 2022 ugh!) But I would replace the radiator, drain and replace the trans fluid, oil, and coolant and see if that makes a difference. Hope this helps!