If you have 12.5 volts at the battery and at the battery connection on the starter and the voltage does not drop when a load is put on the battery, but you only get 5 volts at the starter solenoid connection, your problem is not the battery or the starter. You have a problem or problems in the wiring harness. You need to get out a wiring diagram for your car and trace the starter circuit from the battery through the key switch all the way to the starter, and clean/tighten EVERY electrical connection between the battery and the starter solenoid. If you find any crimp type butt connectors anywhere in your car's electrical system, cut them out and solder and shrink wrap the connections. If there are any relays between the battery and the starter solenoid, you should probably replace them.
A trick you can do to help keep the problem from reoccuring is to install a relay between the battery and the starter solenoid connection. Relays will pull in with less voltage than is required to pull in the starter solenoid. I suspect it is due to a smaller coil and lighter spring on found in a relay compared to the spring used in the bendix of a starter solenoid. Get a Bosche type relay 20-30 amp capability across the points should be good. There will probably be 5 terminals on the relay, and most will have a diagram printed or embossed on the case to show which terminal does what. 1 terminal will go to the coil, 1 will go to the ground side of the coil, 1 will be the common side of the points, 1 will be normally open(n.o.) and 1 will be normally closed (n.c.). If you get a relay with a mounting bracket or tab on it, you can mount it to an existing screw under the hood or with a sheet metal screw in an unobtrusive location. Run a wire directly from the battery positive terminal to the common terminal with an inline fuse holder with a fuse rated equal to or higher than the rating for the points of the relay. Run another wire from the n.o. terminal to the solenoid terminal on the starter. Next run the factory wire that energizes starter solenoid to the positive side of the relay coil, and run a wire from the negative side of the coil to a frame or engine ground. When your starter is wired this way, the key switch will energise the relay and you will always get full battery voltage to the starter solenoid.