You know to be careful when relying on the internet for legal advice, which this is not.
New Your City: One site informs me that "shotguns may be kept in the city for only 24hours while in transit and these must be unloaded and stored in a locked container or automobile trunk
for the period.
District of Columbia: The same site provides that "District of Columbia laws prohibits the transportation of firearms through the city unless the travel is to or from lawful recreational firearm-related activity
The general rule it references is set out on that site, but you will note that a firearm must be unloaded, cased and locked in the automobile trunk
. Once you reach your destination municipal laws may come into play.
Please do NOT rely on the above or the web site, which is located at http://www.cjrpc.com/interstate_1.htm
. It is quite a few years out of date and I note it is wrong in at least one respect. If it is wrong about National Parks it may be incorrect as to other laws. One thing that is clear to me is that, if the laws are still the same as referenced above, transporting your shotgun in the manner described would at least be in violation of laws of the District of Columbia, New York City, and the so called general rule. Re that generl rule, I believe it incorrect as to the transportation of an unloaded shotgun in a motor vehicle (long guns need not be in a locked trunk). Of course I may be wrong about that one.
You didn't mention, but a lot of frequent business travelers stay in hotels and motels. Some take their security firearms into the room with them. That raises all sorts of different issues depending on the jurisdiction.
See if you can get a pamphlet, possibly on line, for each jurisdiction you intend to visit. The info you seek may be available from the NRA or an online source such as Calguns.net (of course for the particular local).