I don't know the current Regs but I would say Yes, no problem for you to hunt there in a DIY situation.
You can look up the AZ regs and also call them.
http://www.azgfd.gov/h_f/hunting_rules.shtml US laws are different than Canada as in you don't need a guide (like canada) to hunt the US if you are not a US citizen.
You would get an out of state license and and over the counter tag or put in for a draw only (LEH) tag.
Each State has different laws but if you have your Hunter number for Canada, etc you should be able to obtain a hunting license in the states, no problem.
I moved from the US to Canada - BC and all my hunting + firearms permits were transferable from the US to BC.
Camping: We camped on public / state land and it was free.
PS: My first bow kill was with a recurve was a Javelina in AZ.
The 10 day Javelina hunt was such a great time in so many ways.
Beautiful country. Glad I had a GPS...