Can you really do ternery computing, though? My understanding is that binary is easy because of the small number of logic gates. You'll have to design a lot of new ones for ternery.
The main reason to use ternary computing is to take advantage of
balanced ternary, usually represented as "-1, 0, +1". If you are talking about DC voltages in analog circuits, this translates very easily to "-voltage, no voltage, +voltage'; and has some neat advantages. With somewhat more work, you can use it with mechanical shaft/gear logic, with "counter-clockwise, stopped, clockwise" rotation; however, the inherent "cancelling" of the -1 and +1 which is one reason to use ternary in the first place tends to be hard on physical gears, loosing most of the advantage. There have been proposals to use polarization states in ternary optical computing, but that has other issues.
However, the most obvious fundamental logic state set in DF cart logic is "no cart, cart" which is a natural mapping to "0, 1" binary bit. A more complex design could use "north-heading cart, stationary or no cart, south-heading cart" as a trit, or if you zoom out slightly and consider the storage loop the basic memory cell you could have "counter-clockwise cart, stopped cart, clockwise cart" as your trit; something similar has been hypothesized for superconducting loop storage.
Of course, a true Dwarven computing device would be in base 8 (0,1,2,3,4,5,6,7), as every dwarf knows that that's the fundamental way the universe works. An... octary? computing device would be a true megaproject! More realistically, however, four binary bits stage easily up to base 8 (a nibble), while it would be much harder to do that with ternary. Aiming for a one octal digit (4 bits) adder is probably a good starting point for binary computing cart logic designs, and then working up from there (carry bit, multiple digits, subtraction, and so on).