They are likely going dormant. Tulips have a very brief period of foliage and an even briefer period of bloom. Then they go to sleep for late spring, all of summer, all of autumn, and all of winter. Then if you are lucky, very lucky, they come back the following year for a brief period. Often they don't come back at all. This is the nature of tulips. Stores that sell potted tulips in bloom should explain that.

I don't know where the 'don't water' advice came from. When you transplant growing plants of any kind they need watering, even if when established they are drought resistant. When first moved their roots are not yet settled enough to seek their own water in the soil. They need help and bulbs moved 'in the green' are no different. Your tulips need a drink. They will certainly go dormant soon but that should be a natural seasonal process, not a reaction to thirst.

Watering anything outdoors has not been a requirement for western Washington this spring - we are more than double (in some cases triple) average year to date rainfall totals and have just completed the 4th straight month of record setting rainfall. I doubt it is likely these bulbs are lacking adequate hydration :-)

Once the foliage starts going off color, root growth is pretty much halted for the season so watering becomes much less of an issue. I would not say it is critical with regards to the bulbs in question - that is exactly what spent tulips look like right now in this area. And you do want to be very sparse with water during the summer as well - tulips like it quite dry during dormancy. FWIW, I never water potted spring bulbs once they are done flowering - I just put the pots outdoors until the foliage has died off completely and they receive only what water Nature provides. Once the foliage is gone, I tuck them in the garden where I want them or sometimes wait until fall. If I wait, they get left in the pot virtually untouched.

The OP is in WA but we don't know where. I'm not suggesting tulips need watering at all when they are dying down naturally. But the OP's look to me as if they were too dry straight after transplanting. Rain or no rain. The state of the foliage looks to me like thirst not natural decline. I would only water once, immediately upon moving, and then leave them alone. I don't water when I move my flowering bulbs outside but the OP's ground looks dry to me. I agree there is no point in watering the brown foliaged bulbs now. But the green would benefit imo.