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A lot of the time it's simply cheaper to install a new one. A faucet is a mechanical thing, so eventually it's going to leak.
Isolate the water to that shower, disassemble the handles and take out the stem where the washer is; then replace, reinstall and observe it. When you have a leak, often it may be on the back side, where it actually connects to the wall, in which case you'll have to disassemble the drain work.
When you do change the water heater, try to install a pan under it. If you have water dripping from the shower spout, most of the time the cause is a defective washer or defective seat within the its body.
As long as you have isolation valves to isolate water to that area, it should be a relatively easy fix.
If a lot of water is dripping, call the manufacturer and provide the model number; you may get lucky and find that the product is still under warranty.
When you do change the water heater, try to install a pan under it. The biggest problem is that the lining wears away and you get water dripping from the base.
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The process of dipping steel pipes into molten zinc to galvanize them has been around since the early 19th century, and galvanized pipes still are common in outdoor and industrial water systems.