Good thought! If too short or long it wouldn't pull all the way into park because that's all spring and roller lever but would get all the forward gears -- not sure if short it would reach all the way into reverse though.

Rubber does look good and should make a big difference.

Any auto parts store should have the breaker--autozone also does--just be sure to get the kind with threaded studs instead of the push in tab type. Believe they are all pretty close in size. The breaker and dimmer switch are both fairly cheap and even if not causing the problem today, eventually will and worth replacing.

Just the friction in planetary will allow rotation of wheels with no resistance in air if pawl not catching so that is where that is probably coming from. Pawl is possibly broken or binding and could cause issue and could also be early design where the teeth would wear and round off so as not catch in gear--but would have to be horribly worn, burred or jammed to not engage at all. I think symptoms point to engagement mechanism since pawl apparently isn't going into gear at all rather than slipping out under load.

If pawl OK, then it is engaged by a roller assy which is pinned to the manual lever shaft. When engaging park, it rotates and pushes the pawl against a spring into mesh. It sounds as if not enough rotation is there to push all the way up so it is possible the pin has broken or loosened on shaft. Also possible the pot metal piece has worn or broken so engagement roller that engages the pawl has dropped so it can't push pawl high enough. Worn roller supports were an issue in early units.

When you take the pan off to check, also look above the pawl lever for a small pin extending from the rear of case in lever path. That is supposed to catch the pawl and prevent it from slamming so hard and high into gear it won't release--but very weak and frequently breaks which can lead to not being able to get out of park.

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Maybe be thankful. These cars have a very nasty habit. Seems like all you have to do is park two close together and they breed-- they're worse than rabbits. Pretty soon you listen to a few choruses of "Either those @#$^&* cars go or I do" (or even worse yet, she stays but you and the cars get show the door)--and then you have to make another choice--and need even more money.

Didn't realize yours are the 1/4 ones. Guess I'm used to the V8s and think those are the smaller ones.

If no one has new or used closer, try Mr G's fastenershttp://www.mrgusa.com/fasindex.htm

miscellaneous clips section 45-49---just looking again but can't tell exactly which is discontinued. Think it's the 1/4".

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No relays used if stock but the headlight switch can be removed from dash and then taken apart carefully and cleaned.

Some have reported problems and failures with the dimmer switch and those are a standard item. You should be able to find a universal replacement at Napa or good parts store.

If the lights are going completely out and then back on in that 10 seconds then the circuit breaker is operating -- most likely due to a short. Also very common with that old cloth wire. Check carefully around any holes in sheetmetal the wires might pass through and points where they exit loom.

EDIT: Since you have been working around the trunk, also check carefully the tail light wiring. A short is very possible in that area--either from something hitting it in the corner near the lights or particularly in the area where it goes across the trunk floor to get to right side.

If you have the late revision to the distributor, those values are in tech bulletin 56T-19. The idle speed is a bit high because according to the bulletin the centrifugal advance will be coming in well before that speed so might be throwing your reading off. Packard still recommended the 5 degrees but then went on to mention that it could wind up more or less depending on other factors.

There were some different carbs used for 56 and descriptions and instructions are in Service Counselor 29-10.https://www.packardinfo.com/xoops/html/downloads/SC/SC-VOL29NO10.pdf Maybe one of those might be what you have.