You're right, it did produce FM. So I reworked the modulator stage, taking it away from the oscillator. I used a 6CS6 pentagrid tube (a sharp cutoff relative of the 6BE6) as the modulator. No more FM (or it's greatly reduced). According to my ham radio TS440SAT transceiver, which can be set to AM, FM, USB, LSB and CW modes on every frequency and band. I repurposed the old modulating triode to become a cathode follower to produce unmodulated RF, to feed the "sync" jack.
The revised schematic:
But it doesn't work on the top 2 bands, 15MHz and up, but I'll worry about that later. Actually it does, it's just hard to tune it to a specific frequency. Also I had a 6211 instead of the 12AT7 plugged into the osc circuit.