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Just be carefully using a ballast or other type of wound-wire resistor as the voltage output is based upon amp load. The less the load, the higher the voltage.

Most of the instrument voltage regulators are "Chopper-Type" which keeps a constants voltage regardless of load (with-in rated load). As an example the 1951-54 (6v) cars use the same regulator as the 1955-56 (12V) cars.

Those "Runtz" style voltage step downs use either a special diode or a IC chip and are usually rated at .5 - 1 amp. The ceramic resistor style step downs are usually used for motors and non-voltage sensitive items.

Update: I went over this evening with a fully charged battery and additionally put a battery charger on it as well and the measured voltage at the solenoid was still in the 5.65 to 5.85 range but as I worked with the switch and cable, I could see it dropping to 5.35, then to 5.15 as I heard the solenoid trying to engage. I now believe it is the push-in cable, i.e., it isn't pushing in far enough to allow full engagement as I did hear the solenoid activate at least a couple of times. I have to pull the cable out to go into reverse. Next I'll try to adjust the cable to go in further if that's possible and then try it. Even though the voltage is < 6V at the solenoid, it appeared to engage at least a couple times. Does this sound reasonable? That is, the cable isn't going in far enough to allow activation and it's just sitting there humming and running the battery down? This solenoid was replaced on the car a few years ago and was "new" at the time and there's less than 500 miles driven since then.
(The only item remaining 6 volts on the car is the OD solenoid. A number of years ago I had attempted to put this big resistor on the firewall to drop the voltage to 6V from the 12V system, but it glowed cherry red and I disocunted that idea.

Without knowing your exact setup it is kind of hard to offer suggestions. If all you have is a toggle switch, then going to 5.35 is a lot of drop unless the solenoid is jamming and really pulling current. It kind of makes the switch suspect. If you run a heavy wire directly from battery to solenoid do you still get the drop?

There are repro 12v solenoids that fit the later units but not knowing what you have, they may not have the proper energized length for your unit & are still a bit pricy. It might be worth looking at though as it would eliminate the need for a second battery from the sound of it.

I thought the switch might be suspect and replaced it with one rated up to 50 amps. The wire is 12 gauge I think. I probably should just run a wire from the battery directly as you suggest and see what happens. There are no kinks or pinches in the current wire, but you never know until you try it. I tried to obatin a 12-volt solenoid several years ago but at that time could never find one the correct length. I'd have to look in my old notes, but I do recall this solenoid is opposite most in that it is a puller instead of a pusher (but I may have that reversed).