You seem to be looking at small-signal output impedance, when you really need to know the large signal parameters.
You neglected to mention any power supply voltage or current. Or what load(s) you drive.
However, if you had a clue when you picked 2.2uFd, then you don't plan to drive less than 5K loads with full ~20Hz response. i.e. you don't plan on finding any 600Ω loads (which a simple 6SN7 without transformer won't drive well anyway).
> with a 47k cathode resistor into a 2.2uf cap - i can get an unloaded 5vac
6SN7 with 200V B+ and 100V on the grid, 47K cathode resistor, will pass 100/47K= 2mA. Unloaded and large distortion, it should swing down almost to zero and up to about 180V. After the coupling cap, this would look like +80V/-90V, or around 60VAC. If you are distorting at 5V, something is wrong.
> i loaded it with a 2.2k series resistor and a 560ohm resistor to ground and get a max of about 1vac.
SO the 6SN7 is swinging 7V peak in 2,760Ω, or 2.5mA peak. I deduce you must be putting at least 120VDC across the 47K cathode resistor. 6SN7 saturated plate resistance is around 5K or 10K, or roughly 3 times the 2K8 load, so you must have at least 4 times 7Vpk or 30V static plate-cathode voltage.
> make sure my output stage can handle pretty much any gear i hook it up to
If you really want to drive 600Ω loads to high level, +4dBm nominal (+20dBm full scale), you need 11V peak and 11/600= 18mA peak. With a single tube, it must be biased to 18mA idle and able to swing to 36mA peak. A 6SN7 will just-barely pass 18mA at zero grid bias without exceeding plate dissipation rating, but can't swing higher. Even if it could, it can't swing to zero current without large distortion: we'd want maybe 5mA minimum current. And rounding-up the signal current to 20mA peak, that suggests a 25mA standing current swinging from 5mA to 45mA. 6SN7 won't do that. Not even paralleled or push-pull.
Driving 600Ω directly needs a BIG bottle. 2A3, 6V6/6L6, or one of the TV V-sweep power tubes. (6EM7 would be a natural.) Even then, you don't get a balanced or floating output...
> i have some edcor 10k/600 transformers
That's a 4:1 transformation. If the 600Ω side must swing 11V 18mA peak, the 10K side must swing 44V 4.5mA peak. This is more like what a 6SN7 can do. Since the Edcors are not rated for DC in the primary, we look at resistance-coupled transformer output. It turns out to be a tight design. Assuming B+ is 300V, we can set up about 175V across the 6SN7, about 120V on its grid giving about 125V across the cathode resistor, about 10mA idle current leading to a 12K cathode resistor. Cap-couple this (the 2uFd is maybe adequate) to the 10K:600 iron. Gain is about 1/8 (1/2 in the heavily-loaded CF, 1/4 in the iron). With a 1:10 input transformer, two 6SL7 stages, and no feedback, you have gain to the CF grid of over 10,000, so you should be OK with a 1/8 voltage loss in the CF and iron.
And the true floating output simplifies interfacing.
> the 6SN7 is probably about 500-600 ohms out anyway (the equation is in the other room).
1/Gm, or for a triode, Rp/Mu.
For 6SN7 at 150V 2mA, Gm is around 2.5mA/2V, so 1/Gm is 2V/2.5mA or 800 ohms. At book conditions (90V and 250V at 10mA) Rp is 6K7 or 7K7, Gm is 3,000 or 2,600umhos, Mu is 20. Output impedance is in the 335Ω-385Ω range. The 500-600 guesstimate implies Gm of about 1800umhos, which for 6SN7 happens at 2mA to 4mA which is indeed a popular operating range.
But note that a cathode follower will NOT drive high levels into a load anywhere near its output impedance. Wiring as CF does not change the large-signal relations. Large-signal, we still have plate resistance of ~7K. The electrons have to flow through ~7K! Because of DC current, the maximum-power load is about twice that or 14K. We see that the best-power load for a triode CF is roughly 2*Mu times its small-signal output impedance, or for 6SN7 about 40 times Zout!
It is a broad peak so 5K and 20K can take large power too. But 600Ω will be far down the mis-match slope. Also distortion falls fast on the hi-Z side of the peak, but is high on the low-Z side (one nice thing: THD below clipping does not rise as impedance drops much below Rp, we just clip at lower power).