Shifter burning while car idles ***1981 accord!!! Help!***

So I just installed a new weber carb on my 81 accord and I was adjusting the mixture and idle and the car was consistently running for almost 20 minutes, during which the inside of the cabin started smoking and I went to touch my shifter to engage in my gears and it was almost on fire. Also, now the shifter feels as if it has some sort of up and down play in it and the shifting it super rough through all gears. I took off the boot and a rubber piece from the shifter had melted down and it was scolding hot as well.

What could even cause this? Maybe the cat was clogged and the heat from that made my shifter linkage and everything around it start to melt? I have no clue what to do ... I've only had this car for a month or so. This is the first time I've had it running for longer than 15 minutes.

Not sure what to suggest regarding the cause of your problem, unless the shift linkage was resting on the exhaust or CAT. It should be held above it on rubber hangers with a thin steel heatshield between. (I've had the hanger bolts fall out on one car).
http://carphotos.cardomain.com/ride_..._large.jpg?v=0 '76-Oct'80 & M.Y. '81->
If the rubbers hanging the shift linkage above the exhaust are melted I'd expect LOTS of play! At least for '81 models that shifter linkage concept was retained right through 'till 01 on Civics so you should be able to find similar rubbers, bushes & hangers from many other Honda. I think the gearstick ball-pivot & bush bigger diameter around '88, & some early '80's Honda dabbled with bearings instead of plastic bushes at the bottom of the g-stick like some short-shifters (more prone to rattling).
https://www.hondapartsnow.com/resour...1623E52AA0BB45

I'd guess it was a combination of a lean mixture, missing heat shields and a clogged cat.