DC, it'll really help your ear if you work on transposing while singing rather than playing.

Here's one possible approach: Using an overly-familiar song like Happy Birthday, sing it in your usual key. Then sing in a key higher or lower by a 4th or 5th--whatever is easiest. If you find this extremely difficult, try it up or down just a semi-tone or tone. That may be harder or may be easier. Use what's easiest and play with it until you get accustomed to how it feels. Another week, do the same with another overly-familiar song; for me the next choice would be Frère Jacques, then The Water Is Wide, then The Fox, but use just one verse for now. If singing the whole verse transposed throws you off, use just one line. The most basic musical line I can think of to start with is the one for "nyah nyah-nyah nyah nyah".

If you find you just can't get through a whole verse or even one line, record yourself singing it (a line or a verse), then use a music editor like Audacity to transpose it. (Beware, you'll sound like a chipmunk when it's more than 2 or 3 semitones higher! But that's okay for this application.) Play it back and sing along with yourself till you get used to the new key. Then go back and forth between the original and transposed versions until you can sing both easily; then until you can sing both without the recording.

Eventually you'll start to understand the melody as relationships of distance between the notes instead of a particular pitch followed by another particular pitch.

Any amount of joyful play of this type will help make it easier, so have a laugh with it. Do it with a friend if you can!