We were in this exact situation two months ago and the convenience store was directly across the street!
Yes, there are numerous gas station/convenience stores within walking distance. There are two that are very close. One of them is directly across the street and the other on the side of the building. Depending on the day you are there, that specific hotel (My favorite place to stay while on Business) provides a free meal and drinks, alcohol included, from 5-7pm Tues-Thurs. They also have a little store stocked with a bunch of goodies that is open 24/7, which is located immediately to your left when you walk into the main hotel doors.
The shuttle driver took us to the grocery store while we were there.
A small walk up the street has one. Front desk can assist with exact directions
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