I have a 85 22re that seems to be running off the battery. If battery is fully charged it starts fine and continues to start for a few days but eventually gets weak and won’t start. You can jump it or push start it and it will fire right up. Otherwise, the dash lights come on but it is too weak to engage the starter. I just put a second new (remanufactured) alternator on it and it didn’t fix the problem. The battery reads 12v when off, and around 13.4v when the truck is running. I also just replaced the battery.

 

˟˟˟˟ starter system design.. (Engineering cuts, bit hey he saved Toyota a couple million in manufacturer costs) the power to the starter solenoid flows through the reed switch on the back of the key cylinder(ignition switch). Something like 3meters of under spec wire and a too small switch to handle the current combined with a few decades of carbon buildup and high intensity arcing result in a weak starter solenoid signal.

​​​​​​@Rad4runner, thingy link...​​​

Short version.. If you toss a 1/4 spade connector and 3' length of wire between the starter solenoid trigger and the battery +, every time you touch the wire to the battery you get a good hard starter engagement.. (Yep no safety disclaimer, don't run over your foot because you had it in gear. Oh never mind there was a minor safety warning..)

Yeah uhm essentially this is hot wiring your starter and could lead to bad things, it takes about $1 of bits and five minutes of work to check.

Ray (Rad4runner) has posted very detailed posts about this on every major Toyota forum. :Google:

 

What??

 

It sounds to me like you may have a parasitic drain. Are your battery terminals clean and tight? Bad connections can cause all kinds of weird problems.
The good news is if you're reading 13.4v while running, your alternator seems to be working. Do you have access to a load tester? That will tell you if your battery is bad or not. If none of that show a problem you can do a parasitic drain test, plenty of YouTube videos showing how that's done.