Jazz Lead Sheets - non concert players

I entered my melody and lyrics into the “lead sheet template” in concert pitch. I also see the alto saxophone
as part of the template which is properly transposed and I can view the part from the drop-down part menu.

When I add a player, the resulting score is just rests. How do I “assign” the melody to this new instrument/player and have it be
properly transposed as the alto sax from the template already is?

You would need to copy and paste the melody to this new player in Write mode. If you do this in Full score view, it’s easy to see you have it all pasted and lined up in the correct place.
Edit menu, Filter, use this to filter what you want (or not want).
Edit menu, there are the two options for viewing in either concert or transposed.