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Heathkit RF1U signal generator.
You have done so well, it seems a pity to give up now, when you have identified the fault areas.
1. Mains input. Solved - bad soldering.
2. B range not oscillating. As you have checked the switch contacts, it only leaves the coil to check. Disconnect two of the three ends and measure the resistance. The tapping should be between one third and one half of the total.
3. No RF output. Set the RF range to lowest frequency, say 200khz, and get your oscilloscope out.
3A. Check input to g1 (pin 2) of Output/Modulation pentode, V2B ECF80, with MOD off. Scope timebase set for high frequency (same as when checking the oscillator output, V1 12AT7).
3B. AF input to Modulator valve. Set timebase for AF low frequency. Check at junction R9/R10 for about one third AF output, and at top of R9/g1 (pin2) for more attenuated version. With MOD on.
This will be fuzzy and if you speed up the oscilloscope timebase to settings in 3A you will see the RF as well.

4. Output to attenuator. It is not unknown! for the resistors to be burnt out, due to being connected to the anode of an IF stage without using a 100pf series capacitor.
So check with your DMM the resistance of the variable output pot, R13 3k or 5k. It should be smooth and linear. And inspect the resistors at the attenuator switch - look for signs of scorching will be quickest.
5. Then look for RF output with the scope,set for hig frequency, at 200 khz, no MOD. Work from anode of ECF80 (pin6); output side of C25-R13 variable output; R13 slider into attenuator switching. These voltages will all be small 50/100mv.
6. Check the modulation if above all resolved. MOD on, and look at the anode of the ECF80 (pin6) with scope timebase set for low frequency,AF. You should see the AF envelope all filled in and fuzzy with the RF.

Note: The DMM on AC ranges will be fine for 50hz, doubtful for AF 1000 hz, useless for RF 200khz. Same for an Avo.
You don't say what your scope is. Almost any will be suitable for this. Ignore the siren voices who say you need a 100Mhz digital costing hundreds of pounds. Impossible to "drive" for this sort of job. A good old fashioned Cossor 1039, Telequipment S32 or S51, Metrovick CT52 or Heathkit OS2 or IO12 will be ideal, as designed for this sort of job in testing radio and B&W TV. and very straightforward to repair if necessary. Often turn up on ebay. Manuals on the web or here on the Forum.
And buying another signal generator may get you into the same sort of faults, while all your work to date will be wasted.

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