The Conejos is great, and the hatches should be going strong higher up towards Platoro. We plan to be in the area (based at Trujillo Meadows) July 4th week ourselves.
You're going by the San Juan (northwest NM) and you really ought to fish it. 4 miles of quality water with lots of fish topping 20". I'd recommend hiring a guide for your first time out. See the San Juan River Forum
for more specific info on the river. There are people here who can recommend guides as well. Early July is typically when the monsoons start in earnest and that brings out the ants, which drives the fish crazy. That happened around July 5-6 last year.
The Animas has public waters from a long stretch in Durango proper with public sections on up to Silverton. Local shops in DRO can tell you where and where not to fish.