First, try another cartridge. It doesn't matter what you buy, you just want to use it to confirm if the tape slowing is too much tension in the tape reel pack. Grab something country or whatever you find cheapest. Try and find something with the felt spring pads instead of the foam pressure pad and all you'll have to do to it is change the foil splice before running it. You don't care what the music is, you just want it for testing.
Regardless, if the foam pressure pad has collapsed or disintegrated in the cart you're now using, it needs to be replaced. It won't cause the tape to slow, but it can make it sound like *hit if the pad had gone bad.
I cannot stress this point enough: the foil tape splice is
NOT to be trusted if it's the original. The adhesive goes brittle with age and when you least expecting it, it will separate. The way the 8 track player feeds tape means hat when the splice breaks, it will just continue to feed tape. Since it's no longer going back into the cart onto the "endless reel", it will feed into the machine until enough gets in there that it locks up the capstan and stalls the machine. Then it's a royal mess to get out of the machine and your tape will, in all likelihood, be ruined. Tape splice replacement is the very first thing to be done on any old 8 track tape. And since all 8 track tapes are old now, they all need to be done.
If your cart is found to have too much tension, the only way to "relieve" it is to open the cart, cut the tape splice, take approx 3-5 wraps off the reel and redo the tape splice. Then you have to manually turn the reel to take up the slack again. This takes the excessive tension out of the tape. There is not other way to do it. You have to open the cart and cut the splice.
Once you have eliminated the cart tape pack tension as a reason for the player slowing, you start looking at the machine. I haven't looked at the manual for your model, but the 8 Track players that had DC motors are prone to motor failure. The way it usually presents is the motor will run fine for a period and then start to slow, eventually stopping completely. Let it cool and it will run again and eventually start to slow. I find this is typically excess carbon dust on the commutator (from the motor brushes) in the motor and dry end bearings/bushings. I take them apart and clean/lubricate the motor and it usually returns to normal operation. If it doesn't, the motor is toast and needs a replacement. good luck i that's the case because finding something that works properly in a speed controlled Dc motor8 Track is a nightmare. There are a few custom speed controlled DC motor packages out there, but getting one is again, difficult.
The DC motors with a mechanical governor are a bigger nightmare. Most often, those are just replaced with the electronic speed control motors.
Barry's 8 track used to sell a universal motor (with speed controller) replacement, but the page on his site where he was selling them has been gone for several years now so he may not offer them at all on anything but his full rebuilds now.
If it has an AC motor, it won't have a speed control potentiometer. It will either be locked to the AC mains freq of the grid or it will be mechanically governed. If an AC motor is dying on you, it's pretty much done. I think Barry's 8 track replaces dead AC motors with his DC replacement, but don't quote me on that.
If it runs for a while and starts to slow, I doubt it's the belt, but you can try a new one. I would agree that the belt might be the problem if it ran slow all the time, but not so much running fine and then slowing after a period of running.
Have a look here:
https://russellind.com/client/download/PRB_Belt_XRef.pdf and see if your model is listed. Then you can just google or ebay for someone selling the correct belt. It looks like your RE-7070 takes and FRM12.7 belt. That's a pretty common belt for 8 tracks, so someone should have one for sale somewhere.
The cart moving around can be the pressure wheel that hold the cart in, or it might just be a poor design transport to start with. Some of the 8 track players out there were just"cheap to start with and hence the "pcak of matches fix". It worked because the transport was poorly designed and allowed the cart to move around too much in the slot, which messes up the pinch roller/capstan pressures and the head to tape/track alignments. Have a google search and see if anyone has a service manual available for download and then read through it for adjustments to see if there's something that can be done about that. You might have a hard time finding a manual for that one because it's a receiver with an 8 track in it. There's usually not a lot out there on those types of receivers which are often considered "all in ones".
There are so many things that can go wrong in an 8 track that cause poor running it's not funny. Its a combination of just the way the 8 track system is designed and 40+ years of age piled on top of it. You just have to work systematically and go from one possible source to another until you get it all sorted.
And even if you get one playing absolutely
perfect, all the factory tapes are experiencing "Print through" to some extent or another by now. So you'll always have a bit of a "ghost track" playing in the background, which makes you think the head alignment is still "off" somehow.
8 racks are fun to mess with, but not for the amateur. You'll find you will know more about them than you ever wanted to before you get yours fully sorted.
I just finished going through my "new to me" Wollensack 8075 and Superscope (Designed by Marantz) TDR-830:
One 8 track player is never enough. You will find you quickly accumulate them if you get into them. I've got something like 5 or 6 of them around now. Although the Wolly and the TDR-830 is my "endgame" for them.
You can pick them up for a song still and they're just fun to mess around with. They'll never really be serious audio gear (although the Wollensak 8075 comes about as close to it as they will get), but they're just good, cheap fun.
You just gotta learn to love them, warts and all. "Quirky" is the PC word for them these days....LOL!
Most of the 8 track "suckers for punishment" tend to hang out around the "Tape" subforum if you're looking for more info.
Good luck.