Actually what I was going to try was use a 12aY7 in v1 and a 12aX7 in V2. So a 12aY and 12aX would be in parallel. ...
The traditional way is give each triode its own plate resistor and its own cathode resistor, have a coupling cap from each plate, then a series resistor from each coupling cap to a common mixing point. Each triode is then 100% independent of the other.
If you share the plate load, you can mix each triode's output together at that shared plate load, but the triodes are semi-dependent on each other: plate current variation of one alters plate voltage of the other, and the load must be the same for both (can't have different-load for working a different part of each triode's curves).
Shared cathode resistor, as noted before, causes signal-input to one triode to create a signal input to the other triode due to the shared cathode connection. r you add a bypass cap to eliminate that, but now both triodes share the exact same bias & voicing. That might reduce the value of having 2 triodes, and it seems like you should just strap the grids together while you're at it.
I don't know if your original setup is the cause for oscillation (never tried that particular connection), but it seems logical that the triodes were fighting you trying to make them amplify differently (with volume controls) when they were tied tightly together in every other way.
Perhaps you might try completely separate triodes first with the mixing resistors (see the 220kΩ channel mix resistors in a vintage Fender 2-channel amp, right before the channels go into the phase inverter). Get your voicing, etc sorted such that you like it. Then perhaps move to more & more shared parts, if feasible. At that point you can figure out where it ceases being workable to combine circuit elements.