Per TOS, the shuttlecraft is ion powered that leaves an exhaust of antimatter residue. It has a limited fuel/power reserve, i.e. fuel tank(s) and fuel lines which might be magnetic containment in nature. ("Warp core" is TNG, so, not applicable in TOS.) I like to think that the fuel is some sort of "intermixed" M/AM ion plasma (with a little dilithium dark magic holding the plasma together without reacting) refueled by the Enterprise. Insert fuel into the one onboard reactor and you get general power for the shuttlecraft. Also, feed the fuel directly into the two warp engines (the two big obvious cylinders) for warp drive. Two separate systems powered by the same fuel explains the shuttlecraft power/warp drive best: one M/AM reactor; and two M/AM warp engines.
In
The Galileo Seven, Scotty extracted a fuel substitute from the phasers (a very small amount of it) that would run the ship's reactor getting antigravity impulse drive, charge the batteries and other general power, but it was not compatible with the warp engines, so, no warp drive. YMMV

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As for overcoming the huge power requirements for transporters and warp drive, I think the "transtator" technology goes a long way in circumventing the power issues. Spock: "The transtator is the basis for every important piece of equipment that we have."