I guess a lot is going to depend on when you plan your trip.
Looks to me like you stand a very good chance of having water in Arizona in mid-March:
and a very good chance of having water in the Chama in mid/late April:
I second the San Juan as an excellent option. With around 5 days I would recommend sticking with the upper and doing a layover day or two with some hiking and exploring. Chinle is a particularly good spot...