It still sounds like a battery, even though its 3 months old. Its the sound commonly made when the battery doesn't get enough power to the relay. The relay closes but when the extra load drops the voltage the coil releases, this stops the load which closes the relay then opens again, your description of a jack hammer is pretty close.
The battery cables could also be the issue. Where the wire goes into the lug, the copper can get a black coating known as black death, check both ends of the cable. Its insulating and over time it will stop power getting from the wire to the lug. It could be a bad ground at the glow relay, you can try running a jumper cable from the relay back to the battery to check it.
One other thing you can try, get a good set of jumper cables. Hook the negative to the side of the engine block. Then the positive hook onto the large nut on the starter. Use a screwdriver to jump from the positive clip to the lug on the starter to see if it spins. This will confirm a bad cable. If it starts, or spins, remove the negative cable, see if it still works, if so its a problem with your positive power cable.
Hopefully that gives you somewhere to start.