Hi Brad, my name is Ed. Welcome to Just Answer!.
It appears that your Central Timer Module (CTM for short) isn't awake which could simply be a failure on its part, but also could be rodent damage. CTM has many duties on the 2001 Dakota that include everything you've noticed, along with headlamp and dome lamp operation and that nifty chime that you get when the keys are left in the truck and a door is opened. While the 4x4 lamp is actually controlled by the transfer case module (XCM), loss of CTM capability always turns this lamp on as well.
From what I can tell, it's probably a bad CTM unless wiring damage under the hood is extensive. I'll describe some circuits for you to test and then you'll have a better idea of what's going on. Even though you've checked fuses already, I'll describe the locations of the involved feeds since they come from two different fuse block locations.
You'll find the CTM behind the left kick panel inside the cab, which is just to the left of the emergency brake
pedal. Pull the left side sill plate up first and then pull the kick panel directly rearward. Both panels are held in place with clips only. The black 6x6 box screwed to the body is your CTM.
It appears CTM gets FOUR power sources and THREE grounds on its three connectors.
The C1 connector consists of 26 pins and is fed by two key-on power sources from Fuses 5 (20a) and 11 (10a)of the Junction Block (your INSIDE fuse box). The connector is numbered to be read from left to right as you look into the open (terminal) end of the connector and the securing clasp is up, with Pin 1 being the top-left and 26 being the bottom-right terminals. Pin locations 25 (dark blue/ white wire) and 26 (dark blue) are these power feeds.
Grounds are all located on the C1 at Pins 1 (black/ light green), 2 (black) and 16 (black/ light green).
The C3 connector will be at the opposite end of the CTM and has 12 pin locations, read the same way as C1.
Pin 1 (red/ tan) and Pin 6 (red/ light green) are shown to be direct battery
feeds, coming from the UNDERHOOD fuse block, called the PDC, at Fuses 3 and 12 (both 20 amp mini fuses).
If your powers and grounds are OK, it looks bad for the CTM. While a bad XCM can cause your 4x4 lamp to come on, it won't affect operation of the systems you've noticed that are on the fritz... but a bad CTM can cause the whole lot. Sometimes a simple CTM disconnect of 30 seconds or so will return everything to normal, but it doesn't mean the problem is gone... it's really only a good diagnostic tool and then if it works you replace the CTM anyway. =/
Replacement of the CTM is pretty much plug-n-play and everything will work again except for RKE and Factory Theft (if equipped). A trip to the dealer for programming of the key fobs and activation of Theft will be needed for those functions.