When sending my final grades to one professional school, my transcript shows I am enrolled in another program
Does my unofficial transcript reflect how my official transcript will appear?

Only your university knows how they handle the two types of transcripts.  But it's possible.

If it does list Program A, will Program B regard this as an issue? Or can I just submit my transcript as-is with no explanation?

I think it would be reasonable to inform Program B that this is an error and that you have not enrolled in any program at A for the fall semester.
It's true that this might reveal that you have applied and been accepted to A, but I don't think that's a serious concern.  Every university knows that applicants apply to many different programs.
If anything, the fact that you've been accepted to another program would likely motivate B to try harder to recruit you, since it's confirmation that you are well qualified for professional study, and it shows B they can't expect you to accept a low financial offer out of desperation.

If this is an issue, is there a way I can circumvent it?

Sure; contact the office at A responsible for maintaining academic records (e.g. the registrar). Inform them that your transcript erroneously shows you as enrolled in the program, and ask for it to be corrected.