35 days to do what is probably 3400-3600 miles is really pushing it for this time of year. Can you do at least 100 mile days? With no days off? With short daylight hours? With possible weather delays the last few days?
Remember that you need to eat, sleep, shop, wash your clothes, make camp, break camp, and other stuff. Consider shortening the trip, leaving sooner, or being open to using the bus or train at the end. To me starting in El Paso or Houston or something would be better than having to catch a train in the end. Leaving in a few days would be even better.
I am hoping you can make this work, but fear that you are allowing too little time.