That's a long way and therefore, a lot of time. The cheapest panniers might break or tear so you might need to spend a bit more. Furthermore, you might need to think about waterproofing at least for some of your gear because although you can plan to start and end a week long tour during a dry week but you can't plan a completely dry trans-US tour. You don't need to buy waterproof panniers (which are more expensive) and handlebar bag because you could just pack the stuff you need kept dry into dry bags inside non-waterproof panniers and that stuff will/should stay dry. However, a lot of people keep electronics and valuables in handlebar bags (cellphone, camera, wallet, etc) so that might be a good reason to consider a waterproof handlebar bag (although a small dry bag would also work too).

The advantage of good quality (and not the least expensive) panniers is that if you don't feel you need them after this trip, you could sell them for a decent price and likely the "capital loss" would be less than had you purchased cheap panniers (and not been able to sell them).

Just my 2¢