Cassette Adapter Fast Forwards During Connection

Your cassette adapter is p'bly acting normally -- you're just able to hear the noise it makes during quiet times. In order for the cassette deck to operate properly, it has to sense that the hubs are moving (otherwise it's gonna think it's reached the end of a side and must reverse play direction; if the hubs never move it'll do nothing but reverse-reverse-reverse).

But the hubs want to turn a lot faster than they need to; the capstan & roller move the tape at proper speed, and the hubs turn harder in order to keep tension on the tape so it spools tightly. There's some gears in the adapter that attempt to let the hubs turn but not too fast to be annoying.

But we're not talking Swiss watches here...sometimes the parts don't mate up well, or some parts are missing, and the hubs do indeed move at "fast forward" speed. With no music playing, try hitting the fast forward button -- any change in sound? If so, your adapter is p'bly doing as well as it can. I went through several new adapters (two of which were missing parts ;-) before settling on a decades-old one laying around in my junk drawer; yes it makes some noise, but not enough to be audible over normal road noise.

I'm sure it's the tape actually fast forwarding and not the normal noise because the stereo screen displays the "FF", instead of "PLAY". I just can't seem to figure out why it would fast forward when the music being played reaches low volumes.

I did however notice that it happens more when i turn the car stereo's volume knob up to hear a softer song.

It's been awhile (Gosh that's ancient!) but some tape decks have a FF on silent feature (designed to fast skip to end of side A and and quickly switch to side B). You may want to dig up the manual and read on how to disable this.