How can the DoD fix the 'spend your whole budget' approach that units have?

Same way they do in the private industry, trough bonuses for budget control. You go under, bonus. You go over, no bonus. Both on a graduated basis. If you miss your budget by more than 10% either way and your budgeting skills come into question. Another, is the use of zero based budgets. You have no budget and have to prove the need for each line item.

I believe this has everything to do with how funds are appropriated by Congress where they must be spent in that fiscal year. Thus, the "hurry up and spend" that comes in early September. I sure wish there was a way that a commander could bank the funds for use at a later day when they might need them. That would encourage savings and thriftiness. Then again, some commanders would get really cheap and probably not buy what they need now in order to save. I do believe that this is literally an act of Congress required to change it.

This is a curse of the Appropriations Act. That law sets up certain categories of funds that can only be used for certain purposes. These funds have expiration date. The one you are mainly referring to is Operations and Maintenance (O&M or OMA) which only has a one year shelf life on it. If you don't use the funds, they get returned to the treasury and most likely you will get less next year. The only way to change this is to change the law.