When the water is running, some of the hot water comes out of the tub spout, even though the shower is selected.

How much water is coming from the tub spout? Are we talking 20% of the shower volume? More? Less?

How is your water pressure when taking a shower?

After turning off the water, the tub spout still leaks/drips.

How long after the shower is turned off does the water continue to leak? All the time or just for a few minutes?

>How much water is coming from the tub spout? Are we talking 20% of the shower volume? More? Less?

I'd ballpark and say 5%-10%. It's like a trickle of (very!) hot water.

>How long after the shower is turned off does the water continue to leak? All the time or just for a few minutes?

Hour(s). Sometimes it is a stream. Sometimes it is just drops.

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[(https://www.houzz.com/discussions/will-my-attempt-to-fix-this-leak-end-in-disaster-dsvw-vd~3783787)

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[(https://www.houzz.com/discussions/will-my-attempt-to-fix-this-leak-end-in-disaster-dsvw-vd~3783787)

The best thing to do is removed cartridges and take them to a plumbing supply house to see if you can get parts. I found parts for a 47-year-old faucet, so there is certainly hope for yours. Take lots of pictures (or video) while you take things apart and label everything with a number and location so you can put it back together.

In what situations would this end so poorly that I need to remove tiles and completely replace the fixtures?

If you destroy something while taking things apart. OR, if there are no parts to be had, it may become necessary to completely replace the fixtures - sometimes old plumbing parts may be held together partially by scale and won't go back together, or will leak worse when put back together. The reality, though, is that you have to take care of this before it turns into a damage-causing leak in your wall.

If this fixture was installed at the time the house was built (1959) more than likely it is of the pre-cartridge era and uses discrete parts like washers and seats and such.

I'd look around for an older, experienced plumber from the area and invest in a one hour minimum charge to get him/her out to see what you have and give you a concise estimate. If the plumber is from the area and has been around for a while more than likely they know this valve and know what fails and know if there are parts available. The cost of that one hour would be the best money you will spend BEFORE something is screwed up and costs you much, much more.

If you're in an urban area you might check the union hall for advice in finding a plumber (working or retired) that is intimately familiar with that era of construction and plumbing hardware.

The fixtures are in remarkable condition which lead's me to think they will not be difficult to disassemble. I take it you own and know your way around tools so here we go. First off since you are only having a little by-passing at diverter,let's only do it to see how things go. The worst that can happen unless you get crazy is you make the diverter worse but there's no flood and surly you can take tub baths for a few days without drowning. Set handle at mid way between tub and shower. Hold knob while removing screw. If handle doesn't slide off with a little pulling and wiggling,pull while tapping with metal object from behind. After handle stem cover are removed you will likly see one small plus one large place for a wrench. Don't mess with small nut,that's to adjust packing in case water leaks around stem. Use a box end wrench or deep socket (6 point preferred but certainly not nessary)to remove stem. O-rings and washer are self explainitory but seat that remains in fixture can be tricky to remove and difficult to find replacement for. If you don't see shinny scratches on seat you might get by without replacment. The seat will have either square or hex hole in center. Seat wrenches are inexpensive but if you are carful,seat can be removed with ever day tools. Allen wrench for hex and an old manual operated wood drill bit for square. Just make sure it fit's and keep it straight or you risk damage to new soft brass seat while installing. While stem is out,lub stem where it go's through packing plus course threads that pull and push stem when handle is turned. If that go's well you can put new washers in hot and cold before they start dripping ( don't forget to adjustmid-way open and closed while removing and reinstalling). Good luck

When you turn off the shower the water that is running from the tub spout to the shower head loses pressure in the pipe and runs back down to the tub spout, in newer homes and with newer pipes this happens pretty fast. However, in older homes with lots of buildup in the pipes this draining of the shower head pipe can take quite a while. Hours seems a bit weird to me, however, so does a leaky faucet that stops leaking after a while.

I personally think you have buildup in the pipes that is obstructing both the diverter valve and causing the pipe to drain slow. But I absolutely could be wrong.