It seems to me like the teacher was trying to take you back a step. Reading sheet music and writing it out as letters is a great way to improve music reading skills in beginners, or for scraping the rust off! The more you write the notes out as letters, the faster you become at reading music, but you don't need that, you can fluently read sheet music, so I don't see the point. It's probably more for him, that's how he teaches, so if you do that, he'll be able to progress you on to the next step in his teaching routine, basically, he'll know where he's up to in his plan.
As to the problem you have "jamming" and knowing where to come in, perhaps karaoke style backing tracks with a melody guide might be useful, quite a lot have the melody line played quietly on a synthesiser. That would let you know where you're up to in the song.
As for what to play, a simplified melody line is good, you'd easily follow that reading music. Otherwise, you can follow the chord progression, playing scales or the notes of the chords. Guitar sheet music will give you both the chord progression and the melody line, but of course you'll have to transpose either the sheet music or your playing if you're using backing tracks. I like a website called Karaoke Version, there are free backing tracks, or you can pay a couple of dollars for others. Once you pay for a track, you can customise and download it as many times as you like, supposedly "for life". It allows you to adjust the volume of each instrument on the track (the instruments used vary according to the song), to mute the instruments you don't want, and most importantly, adjust the key. So if it suits, every song can be in Bb or Eb to make life easier, or just drop the key so that when you play, the track matches the sheet music so that you don't need to transpose what you're reading. Some backing tracks will have saxophone in them, having that play at a low volume in the mix would give you something to follow and might provide ideas for your own improvisation, very rarely will it be the main melody line.
Hopefully, a visit to Karaoke Version .com will help you to have better tracks to play along with.
The single, most important thing to remember is that this is supposed to be fun, you can't really do it wrong.