Looking one chord or one bar ahead is way too shortsighted. And whatever the next chord is, it really shouldn't be a surprise to you. It may be because you don't know relationships between chords that well. This often happens when people learn song after song: the chords are in that order for that song, but no pattern is perceived between the chords themselves, only the pattern of those chords for each song.
Knowing what the I chord is - the tonic - in a piece, and especially in a key, is paramount. From there, each of the other chords will have a relationship with that I.
V>I is a common enough change, and probably the most common. IV>I is another. By being able to map out the whole sequence of just one well-known song, you should be able to find, for starters, those two changes. Listen to what happens, to the feel of the changes, to what happens when I goes to somewhere else, to what happens when one chord is lingered on for longer. Does it feel like the harmony has moved upwards, downwards, a couple of notes haven't changed, all of them are different in the next chord?
You say jazz piano. Sounds like you need to pare back the chords to simple triads initially, as most jazz chords won't be the simple triads we all know and love! But do that, C13 still works when it's played as its base triad of C E G. Won't sound so jazzy, but will give you more information that you need right now: it's major. So, an idea: re-write one of the songs, taking the chords back to bare bones. That may take a bit of doing with some jazz chords, but when it's done, play simply, and listen to what the changes do. Relate those to other keys, so you start to understand the relationships - C>F is the same as F♯>B, as A&flat>D♭, etc.
And, as John says, don't just look at the next chord, but map out at least full lines, if not eventually the whole thing. But try not to think of it as 'that bit goes C>G>Am>Em. More along the lines of I>V>vi>iii. So you're not thinking key-centrically.