What parts did LBS say were worn out, the shifters/derailleurs , the chainrings /cogs, or both? I would guess at very least the latter were worn out, which would degrade shifting.

The cranks you have will work but you almost certainly need new chainrings in front and casette in back (freehub or freewheel depending on your wheel -- although maybe your replacing the wheels?). By the time you add up cost of three new rings for the crankset, you probably would be spending as much or more than you would for a whole new crankset and BB of similar or even slightly higher quality. So I'd go with new cranks