1. first you need to remove the cranks and determine the reason the puller won't thread in.
a. clean the dust cap holes real well.
b. do the same to the puller threads.
c. apply a grease or light oil to both the puller and crank holes.
d. try again to thread in the puller. go slow and even just using your hand at this point -- no wrench.

did it thread in? if so, go all the way with your wrench. try the other crank too. if it won't thread, don't force it. you may need to correct the threading at this point before moving onto bottom bracket issues.

but about your bb ...

just because a shop said you need a new one doesn't mean as much. maybe all that's needed is an overhaul and readjusting the adjustable cup. tackle all this after removing the cranks.