Steve, I think it is wonderful you have a goal to ride across the country. It is definitely inspiring.
I was hoping you could provide some clarification. I am not trying to be rude; just trying to ensure a given route is feasible for you, so please forgive me if I come across as inconsiderate as I am definitely not wanting to be. I know one person locally who occasionally uses a hand cycle and he must ride with someone or have someone available to come get him in case of a breakdown or other issue so my experience is a bit skewed which is probably not be an accurate situation for you as he only rides maybe a handful of times a year.
So some questions in no particular order: While I read you may do as little as 30 miles per day, how many miles a day can you currently reasonably do on average? How many do you believe you will be able to do after training? At what percent grade does the road become too steep? If it gets too steep, what do you do since I assume you may not be able to walk or if so may not be able to pull/push a 100+ pound trike & gear up a mountain for 5 miles. Are there other medical issues involved that need specific requirements, i.e. electricity, keep meds below 90*, must sleep in a bed, etc.? While you said prefer camping, are you able to do indoor lodging when no camping is available? What are deal-breakers in a route, such as a road grade over X%, elevation issues, camping only, etc.? Are there time limitations, i.e. must be done in a max of 3 months sometime between January and July? Are there any limitations or requirements we should consider (must visit Phoenix, biggest ball of string, etc., no more than X miles between food supplies, etc.)? Where ideally do you hope to end the trip specifically; Lubec, ME, Boston, etc.? How will you get home after the end of the tour as a secondary route to a transportation area may be needed? If you do not have one already, are you able to get a passport as dipping into Canada in places might be better?
Again, not trying to be inconsiderate, just seeking information so we can suggest a viable route for you.
Once you get east of the Rockies, it should be doable. However, before we get your answers, I was thinking about using rail trails when possible due to the easier grades and maybe keep more northern in the northeast since a trike is not always pleasant to ride on a narrow highway with no shoulders and lots of traffic, i.e. the major metro areas of the Northeast might be best to avoid.
Tailwinds, John