It is possible, but your comments are two separate subjects. First, go to http://www.handgunlaw.usand look for the link for "Non-Resident Permits".
That will give you a list of states that will issue one for you and what their requirements are. If yougetUtah AND Florida license, you will be good to go in about 30 states.
A Washington license will be good in about 8 states. That site can give you a map of the states each license is good in. This is called reciprocity.
Utah and Florida will issue through the mail. On costs over $100 the other around $70. Well worth it in combination with a Washington license. I am pretty sure that as a resident of Washington, you will have to have a Washintgon CPL to conceal here. The other two licenses will allow conceal carry in a whole lot of other places.
Hope this gets you started on an answer to your questions.