The other responders have different systems and are only speculating based on what their system does.
The 2008 Winnebago Destination, exactly what I have with the same system components as you, only has two modes of leveler operation. The only way to dump the air bags is by using the levelers. One is the Auto, and the other is Manual. When the Auto mode button is pushed the air will dump and about 20 seconds later the two left side jacks will deploy, then the two right side, and the "auto" levelers will balance out the remaining setup.
If you decide to use the manual mode, as soon as you hit one of the bi-level switches the air will dump. However because it is manual mode the levelers will continue to move. It is best to wait for the dump to complete before continuing with the manual mode. When the system is working correctly I always use Auto, then touch up in the manual mode.
For both modes, and including the slide operations, it is recommended by Winnebago that the engine be running. It is only to help with the current by letting the alternator contribute to the load. There is a lot of current on those slide motors, one for the little slide and two for the big 20+ footer.
I always level first, then the small slide goes out, and then the big slide goes out. Just in my mind the little slide gives some counter-balance for the weight of the big slide.
As for the airbags not dumping, that is an unusual situation. I suspect it could be a sensor thing. I have had the reverse happen where the air bags do not inflate. Talk about a teeth rattling ride when you take off, wow! The remedy for them not inflating is to extend the jacks (level, level ground required) as far as they will go and then retract them. That has worked, recommended by Winnebago, each of the two times it has happened to us. It may be a possibility that it would work in your situation and would not hurt to try it. It may trigger the sensor to work again.
Otherwise - it's to the repair shop.
p.s., I have had a lot of problems with mine over the past 5 years - mostly fixed - so if you have any specific questions, ask away.