[Bob Lincoln]

Sure sounds like there is a fault detection and shutdown going on. There may be a resettable fuse (positive tempco resistor) for protection, which could either be doing its job or be faulty. As you know, there are multiple voltages for VF displays - I recall from my old days that there is an AC voltage of, I think, 4 volts on the filament. The grid is higher, can be as much as 28 volts. I found this website that may help, for automotive:

http://hem.passagen.se/communication/vfd.html

 

[ImperialCrown]

In a darkened area, you may see the horizontal filament lines lit a dull orange when the ign switch is on.
There is a dc-to-DC converter that steps the 12 volts up to ~22-28 volts. It may be packaged like an IC. The MOV looking device may be a regulator or protection. Measure the pins when the unit is first turned on and watch for the voltage to drop out after a moment.
The electrolytics were a known common failure, especially after 18 years of service. 'Freeze spray' aerosol may help to locate intermittants.

 

[B10alia]

As for repair, you may be hard pressed to find any information on the unit in the way of schematics. I was trying to cobble together a portable stereo out of an older GM radio, and, as it turns out, electronics companies/divisions really don't like to release schematics of their products or specs on any of the components. In '95, Chrysler still owned Huntsville, but I still doubt you're going to run into a whole lot of information, unless IC or someone else who has a dealer connection can find something for you. Repair will be in the hundreds of dollars, based on initial quotes for a burnt-out Infinity head unit I tried to get fixed a couple of years ago. Fortunately, the AJ-body travelers were decently common, and you may be able to find one on eBay for less than $40 shipped. I bought a (considerably rarer) AA body traveler on there for $20 shipped recently. That would be the route I would go; I'm pretty inept with electronics repair, but that's just my $0.02

 

[ImperialCrown]

These were never serviced at the dealership level other than replacement or exchange. Our regional authorized service depot (United Radio-Syracuse, NY) were the go-to guys for radio, instrument cluster, modules and other automotive electronic repairs.
This service information and many specialized parts are not for the public and are Chrysler-priority only.
If you are good at electronic repairs, you may not need a schematic. Having it plugged into the car connector with the covers off, may allow you some voltage checks. Compare working with not-working voltages.
Delco for instance would go as far as say, releasing a unique 7-pin Delco chip where a standard 6-pin would be available as a generic equivalent in order to keep the parts and repair business their's alone.