There's the oft-quoted II-V-I, or ii-V-I sequence that occurs frequently in jazz (and lots of other music). Try to find those parts in each song, and once you do, at least the next couple of chords will be in a familiar pattern for you.
Another thing to try, in the initial stages (and for a bit of fun later) is to keep to one key. This way, you'll see the same little sequences far more easily. And it'll make your transposition better, too, both on instrument and on paper.
As Shev says, keep the melody and chord sequence in your head, because at the end of it all, that's what the song is, and both will merge in different ways as you improvise. Again, keeping to one key might give you chance to use specific sets of notes - provided you choose songs that don't modulate much, and so stay in a key longer.
Try just changing the timing of the melody line in some, so the same notes are used - maybe swapping some notes round in given bars. It's pretty safe, and will usually work.