Ummm.
To start, do you have a tube tester? Weeding out a weak tube could save time and aggravation.
From your description, it sounds like a coin toss between a weak crystal and a weak tube.
When a crystal gets old, it will provide less positive feedback to the tube. And that's what makes that stage of the radio an oscillator. Positive feedback.
I would check the 150-Volt 10-Watt zener diode on the underside, just inboard of the final and driver sockets. If it has much less than 135 Volts DC or so on its positive terminal, this can create the problem you're seeing, even if the tubes AND crystals are both okay.
When you select a mode, either AM, USB or LSB, you are also selecting one pair of crystals, one each from two sets of three.
One of this pair is the carrier oscillator for that mode. For LSB, it's CR2 on the schematic at
www.cbtricks.com/radios/browning/golden_eagle_mark_3/graphics/browning_golden_eagle_mark_3_tx_sch.jpgCR2 runs at 5.648 MHz, just above the center frequency of the SSB filter at the center of the radio. This way, the filter blocks this internal carrier frequency, but lets the lower sideband through, since BOTH sidebands are coming into the filter from the balanced modulator inside T1.
If the trimmer capacitor for this crystal (C2) has an oxidation problem, this can shut the crystal down, or prevent you from getting it set on frequency. If turning C2 doesn't change the frequency of CR2, you may need to disassemble the trimmer and clean the oxide from the spring clip underneath and from the end of the pivot shaft where the clip holds it in place. Kinda like rebuilding a Timex wrist watch. Just be gentle to the spring clip that holds the trimcap together.
The second crystal of this pair used in LSB mode is CR5, running at 5.047 MHz. The trimmer cap for CR5 is C34. It can affect this crystal if it has a problem. This crystal is mixed with whichever 16 MHz channel crystal is currently selected, to produce a 21 MHz output from V6B, feeding into L7 and T3.
That is, unless you have the early version, which does not have an ALC trimpot, nor does it have L7.
CR5's output from V1B, feeds into T6. T6 is a VERY narrow-banded step-up transformer. The peak on the slug in T6 is V-E-R-Y sharp!. Sometimes, the setting of the slug in T6 can cause the 5 MHz crystal CR4, 5 or 6 to become "lazy", and fail to start up when you change modes. Just setting T6 to one side or the other of the max-signal peaked position can cause this, too.
Last but not least is the mode selector itself. The contacts will become oxidized and prevent one or both of these crystals from running. If rocking the mode switch just a tiny bit makes your transmit signal drop in and out, this is a clue.
JUST BEWARE THE INFAMOUS "blue plastic" hub found in some of these mode-selector switches. This blue plastic will turn to guacamole if you use a solvent that is too harsh. Alcohol is safe, but the contacts need a thin layer of lubricant to prevent wear. Plain silicone oil won't hurt anything, but many "silicone"-containing spray lubes ALSO contain solvents that can destroy that blue-plastic center hub.
And if your Mark III has the older mode selector made ONLY from the brown resin and metal, you can use about any solvent you want. But if you see BLUE peeking out from around the center hub of the mode-selector sections, be very careful. Ruin that switch, and big trouble follows.
Kinda like "My car runs rough". No way you can pin that down to "just one" likely cause. Too many possibilities.
A really strong tube may hide a weak crystal. And a weak crystal may run fine with one tube, but not another, even if both tubes check okay.
73