The lyrics I googled do not sound bawdy to me at all, but you may double check for double entendres. If gender is the main problem, you may try something like:

Help me, Jane, I'm bully in the alley,
Way, hey, bully in the alley!
Help me, Jane, I'm bully in the alley,
Bully down in Shinbone Al!

Ally is the guy that I love dearly,
Way, hey, bully in the alley.
Ally is the guy that I spliced nearly.
Bully down in Shinbone Al!

Help me, Jane, ...

For seven long years I courted handsome Ally,
Way, hey, bully in the alley.
But all he did was dilly and dally.
Bully down in Shinbone Al!

Help me, Jane, ...

If I ever get back, I'll marry handsome Ally,
Way, hey, bully in the alley.
Have six kids and live in Shinbone Alley.
Bully down in Shinbone Al!