That does sound like an alignment issue (though it's the tape guides that go out of alignment, not the video heads themselves). They can loosen over time. If they are too off, the VCR can't fully align to the video signal on the tape.
There is a short explanation of how to adjust them on some decks
here provided it's a vcr where the guides are easily adjustable (don't know what VCR you got). I recently used it on a samsung deck to play back a tape that was recorded on a misaligned vcr, so it's not just on JVCs you can easily adjust it. There are some images of the tape guides a bit down in
this post.
SLP and LP are more sensitive to tracking problems than SP, so it makes sense that it's more noticeable with those speeds.
Alternatively, if it's a cheap VCR, and you don't want to mess with the alignment, getting another lower-end deck is not too hard. If it's e.g a high-end S-VHS deck that may not be as appealing.