I often say I’m going to the market; sometimes I say grocery store. Occasionally, I’ll say “I have to go down to Martin’s”. I also will say I have to stop by the produce department to pick something up.
Where I grew up just outside of Baltimore, the thing on wheels at the supermarket was a “cart” or “shopping cart” or “grocery cart”. I still say cart, but the regionalism here in western Maryland is “buggy”.
I always say market when I am going shopping. CARE-uh-mell is the pronunciation that everyone I know uses and BM has been the way going doodie has been described since I was a kid.
When I read the OP, I didn’t quite get the problem. 
And heading out of the Bronx it’s the New England Thruway or somesuch. :rolleyes:
Ohio Route 202, U.S. Route 35, Interstate Route 75. These aren’t necessarily the formal names, but they are appropriately descriptive and I do hear them occasionally. It wouldn’t be common to abbreviate the last one to “Route 75,” but it wouldn’t be wrong, as such.
I was brought up in Saskatchewan saying ‘CARE-uh-muhl’, BO is “God, you stink!”, here in Texas a shopping cart is sometimes a ‘basket’, and I get groceries at ‘the store’ because I work there.
In chicago, everyone says “pop.” It clunks in my ear every time someone calls it “soda” on a show set in Chicago.
ER always used “the” in highway designations as well, which bugged me. “The 290” and “the 94,” of course, are really “the Eisenhower” and “the Kennedy” (north of Congress) or “the Dan Ryan” (south of Congress)!!!