Awasezu is rice vinegar + salt + sweetener (usually sugar, can be rice syrup). You use that to season your sushi. (Rarely sushi chefs will add lemon juice to their awasezu.)

Rice vinegar is missing salt and sweetener. You can use it as a flavoring in cooking.

Mirin is rice wine and you don't use it for sushi. It's a flavoring used often in Japanese cooking.

The best way to discover if you're holding the right bottle in the store is first, does it say "seasoned rice vinegar"? Likely you have the right kind, but check the ingredients. I have two bottles (Central Market Classics and Marukan) in my pantry and both list the ingredients as Rice Vinegar, Sugar, and Salt. The Central Market one also has brown sugar and I don't like it as much as the Marukan one.

Supermarkets don't always stock awasezu. For some time I had to buy a few bottles at a time off of Amazon This is what you're looking for.