Hi. My 1983 T-1550 has electric brakes. I can't be sure the T-1350 does. I say this only because the 1350 is a fairly light trailer and may not have required them....esp. in 1983. However is fairly easy to verify. If you crawl under the trailer and find wires coming from the axle tube to the brake drum back plate, then you do have electric brakes. Wither they work or not is a different story.
The "brake box" you refer to is I'm guessing the breakaway cable switch. If you have one mounted on the "A" frame side, be it the right side or the left side or on the top of either somewhere near the tongue that should indicate you have/had electric brakes.
The breakaway brake system is for applying the brakes in the event the trailer comes off the hitch ball while traveling. the cable is attached to the tow vehicle, for example at one of the eyes where the chain would be also attached. When trailer detaches, the cable pulls on the pin which is a wedge holding a switch open. As the pin pulls out the switch contacts come together allowing current to flow from your trailer battery to the trailer brakes causing the brakes to operate. The object is to avoid the trailer from running willy-nilly to who knows where for who knows how long. It is something hopefully you purchased and will never have come into play.
The breaks can also be controlled by a brake controller which is mounted inside the tow vehicle. This works in conjunction with your tow vehicle breaks and trailer brakes. With this setup the tow vehicle doesn't have to do all the work, but allows the trailer to help break itself. The controller can be adjusted for different loads and there are two types of controllers. The caveat is does your tow vehicle have the wiring in place for a controller, i.e. is the vehicle purchased with tow package?
I'm going to let the more experience pick this up as they can be more precise with the systems and how they interact.
In closing I will say one way to think of how the breakaway switch works is when you buy and item that has batteries in it and you pull the tab out for the batteries to make contact for the first time. The items switch could be on but that tab holds the contact open.
TT:1983 Sunline T-1550
TT:1996 Sunline T-2053
TV:2005 Toyota Tundra Double Cab 2UZ-FE i-Force 4.7 L DOHC (MFI) V8 4WD SR5 Automatic
P3 break control
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