The 300 series stainless steel is non heat treatable, so no it can not be annealed.
The 400 series stainless is hear treatable and can be softened or hardened.
There are many options to stainless steel and many series including 700 series.
Inconel, nickel, monel, bronze, nickel bronze, are also to be considered if available check you local welding supplier
sometime they have rolls of stainless that can't be used for welding, for one reason or another, and you can get real cheap..