Since you say the track pump's gauge reads 100 PSI, that suggests the air is pressurising inside the hose. I'd bet that it only takes a couple of pumps to get there too.
This means the valve is not opening to admit air into the tube, AND that the pump head is forming an airtight seal with the valve.
If its a Presta valve, make sure you back off the acorn nut on the tip of the valve.
If its a schrader valve, try depressing the pin in the middle. It could be that the valve seat is not moving when in the rim, though that's unlikely. Its also possible that some sealant has got the valve stuck closed, which would explain your difficulty in deflating the tube.
In this case, try to unscrew the valve core out of the valve, clean it, and then reinstall.
The valve core itself looks something like this:
and the tool to remove it needs to engage the two flats on the left side. You might be able to unscrew it with needle nosed pliers, but the proper tool looks something like this:
The new tube valve is shorter than the old one.
Could be that the valve stem is retreating inside the rim as you try to clamp the pump's head on. I feel this is not your problem, but adding for completeness.
If its Presta, fit the locknut. If its Schrader, then press a thumb hard into the tread of the tyre, outside of where the valve is, to stop it squashing into the rim.
Once the tube's air pressure gets above ~10 PSI it should be plenty to hold the valve in the right place.
Another solution is to use an air pump with a thread-on head not a clamp-on one, if you have access to one. In the same vein, a CO2 based inflater can help, but they're expensive and only worth using if time is of the essence (like in a race or group ride.)