So, what exactly are your symptoms?

Can you shift into the small front sprocket? Is that a double or triple?

Does the chain rub on the inside of the derailleur when shifted to the small front and large rear? In that case, if the cable is slack, then it is a lower limit screw adjustment (or you are beyond the range of the derailleur).

I'm surprised the shop doesn't have barrel adjusters in stock. But, you should be able to adjust the cable tension without one, it just takes a little more effort.