In general, electrical components are either AC or DC. In a TD, there would have to be separate wiring, separate fuses, etc for each system.
This is not 100% accurate.
Assuming you are in the USA the electric fixtures in your teardrop will run off of either 12v or 110v, in your case it doesn't really matter because you know you can power them by plugging into a 110v outlet.
To simplify, the question then becomes can I somehow substitute a 12v battery for the 110v outlet, and the short answer is yes,
you just hook an inverter to the batteries and plug that into the shore power plug

Better yet, get an inverter/charger so you can charge your battery when you plug into shore power.
Take a look at this wiring diagram for a readily available inverter/charger.

- invertercharger.JPG (44.24 KiB) Viewed 3170 times
Unless you want to charge from your tow vehicle you can ignore the connections to the left of (under) the deep cycle battery. As you can see a special switch is only required if you want to change from 110v input to a generator (unless you consider the DC breaker a switch). In your case the shore power inlet would be the AC source panel and whatever that connects to (hopefully a breaker box) would be your AC load panel.
If it is the case that your fixtures run on 12v, you may be better off looking at a converter/charger, but if your TD is coming without a battery I doubt that is the case.
Caveat emptor, to be sure I'm not an electrician. However, I do regularly use and repair circuitry designed to measure tiny electrical signals from the brains of living animals

. Also, I just wired a converter/charger in my tiny trailer to run my 12v system off 110v shore power.