Suggest you check out the official info for Yellowstone
here. Most Park roads aren't open until sometime in May and some campgrounds don't open until June. My suggestion would be to reverse the sequence to head west across TX to AZ and UT and work your way north to arrive in Yellowstone at the end of May to early June.
We have been to Yellowstone and Grand Teton many times over the years and our favorite time is the first two weeks of June. BTW, we always take 8 weeks for the trip, as it's a long way just to get there and get back. Many folks do it in less time, but it's much more relaxed when there is plenty of time.