[old_tv_nut]

1) Yes, the 75-300 ohm transformer is the correct thing.

2) The signal from your modulator should be strong enough to overcome RF radiated interference, even though the 300 ohm connection is not as well shielded as a coax cable. So, this could require some investigation to see how the interference is getting in.

Regarding going via coax directly to the tuner, it maybe doable, but you need to provide a schematic and/or pictures of what's in the set that connects the 300 ohm screw terminals to the tuner coax input.

Questions: Does the set have separate 300 ohm screw terminals for VHF and UHF? Are you sure you are connecting to the right ones?
Are you sure the tuner has a coax input, or could you be looking at the tuner's IF output cable?