Sweet vermouth and red vermouth are synonymous. Either characteristic distinguishes it from dry vermouth, which is clear and, containing little sugar, "dry." As it is fairly low in alcohol, it is usually in the wine section, with the sherries.
If you want to substitute, I think a ruby or tawny "port" or a sweet Marsala would be excellent choices. Cream sherry would also work, but that's starting to get far afield.
After you make the dish, be sure to use your new-found vermouth and make yourself a Manhattan!
Walt