I have been trying to get our firm to go the design/build route for years. Hard to teach an old dog new tricks, right?
Regardless, bidding multiple design/build firms at the outset is a great way to go. Don't be too concerned about the prices being higher in this venture because typically if a design/build firm has a good sub base of contractors they are giving really competitive pricing because they always have that firms work.
One advantage, that I really find of value, is that the design/build process keeps the contractors and architects on the same page. They both have incentive to work together. The more design-bid-build process immediately starts the two parties in an adversarial relationship. Them VS. Us.
One disadvantage to the design/build process is that A/E (Architecture / Engineering) fees can be deceiving. Most contractor led design/build firms say, "If you hire us, we'll throw in the A/E fees because we're building the project." Check this. Review their numbers. It isn't free, but they can hide it in their profit/overhead or general conditions numbers.
The other disadvantage (which may be advantageous to some) is cascading liability disappears in a design/build situation. I find it a disadvantage for the owner, because the design/build company insurance can be harder to pierce than would be in a typical design-bid-build situation should you need to go after one sub contractor.
All in all, both processes will work. But find out which situation best suits your needs. If you have time to project manage, then design-bid-build will be fine. If you don't, and want to hand over project management, then design/build maybe the solution.
Either way, reach out to me if you have more questions. I love talking construction delivery methods.
Good luck!