how to increase the shelf life of bechamel sauce and what is the actual shelf life of white sauce?
You cannot increase the shelf life of random food. There are databases with food lifetimes, if you can't find your food there, assume 3-5 days for any cooked, ready to eat dish - that applies to bechamel too.
You can try freezing it and see if it comes out well; there is a chance it will separate on thawing. If freezing does not work, there is nothing you can do. Adding random things like acids or salt will not increase shelf life. Conservation methods which rely on acid and salt require a certain concentration, which very much changes the taste of the food (think pickles), is not applicable to just any food, and each recipe requires extensive testing before it is declared safe. So you have to accept shelf lives as a fact of life, and work your schedule around them.
See also How long can I store a food in the pantry, refrigerator, or freezer? for a rough overview of shelf life of food types, and visit databases like STillTasty for exact shelf lives.
For the proper storage method, see answers to this question: Storing bechamel sauce.
To improve the shelf life, you can try adding some acid such as lemon. Not sure how long that will allow the flavor to hold together better.