EastHarlem_E, I've lived in and around that border for about a decade. The short answer is that East Williamsburg is a real enough neighborhood in local use, but it's often stretched by brokers. The stretch below Flushing between Graham and, say, Bushwick Ave is mostly Bed-Stuy by official maps and older residents, though some newer renters and listings use East Williamsburg because it sounds closer to the L.
The block-by-block feel changes fast around there. If the place is right on Flushing, you'll get a lot of truck and box-truck noise, especially overnight, because Flushing is a truck route. If it's a few doors south, it can be surprisingly quiet, but check whether the street is full of auto shops and garages. Those bring early morning banging, idling, and occasional sidewalk blocking.
For food, the immediate pockets don't have a full supermarket. There's the Associated on Graham, and a Key Food over on Broadway, but neither is a quick walk from every block below Flushing. Late at night it isn't dangerous to me, but it gets isolated in places because many wholesalers and industrial storefronts roll down at 6. I'd walk it on a weekday at 10 or 11 and see how many bodegas on your route are actually open. Some are, but many are not.
If the broker used East Williamsburg while the lease says Bed-Stuy, don't assume malice, but do ask the management company for the legal building address and the block/lot if anything seems off. Also pay attention to the nearest garbage pickup and whether a bus stop on Flushing means noise and street trash. It's a fine area for some people, but it's less like Williamsburg and more like a quiet-to-industrial edge of Bed-Stuy.