It's not clear from the question whether your fingering is for your own playing, or whether you want to add fingering to the score for other people to play.
For professionally published scores, usually there is no fingering, or the fingering is added by the editor, not by the composer. So if you are "a bit of a novice", don't put any fingering at all in the score.
If you want to play this ultra-legato, at a moderate tempo, without any pedal, a good player could cross one finger over another, and do either 3/1 4/2 3/1 4/2 3/1 or 3/1 4/2 5/1 4/2 3/1, without the finger change.
But piano music isn't usually played like that. It's important that the top line of notes are legato, but the bottom line doesn't matter so much. Most players would do something like 2/1 3/1 4/2 5/3 4/2. Nobody will notice the slight gap caused by the "1 1" on the E and F#.
At your tempo, there would be no problem for a good player to use the sustain pedal to do "legato pedalling," and finger the notes any way they liked.