The RO may have two problems. One is weak output drive.
The other is that loading its output kicks that current into
the supply / ground rails where it can destabilize, mode-
shift or quench the oscillation.
I recommend using a resistor divider network to get a
scaled, 50-ohm output (like, 4950 series and 100 ohms
to VDD and VSS in shunt). Sure, you're now looking at
10mV rather than 1V, but at least it ought to be clean.
Can MOSFET be used in some configuration to match the output to 50 Ohms? Is it a impedance matching problem?
Your mosfets are turning on sufficiently to create voltage swings on a high impedance load.
You need to turn on your mosfets to a greater extent, meaning their On-resistance needs to go lower. Then they can drive a low impedance load.
Your circuit has some output impedance which produces maximum power in a load. By experiment you can find the load resistance. It might be 500 ohms, or 50 k, or 5M. It will draw some midway voltage, at some midway Amperes. The goal is to get maximum Watts.
Once you find out what this impedance is, you might contrive a transformer, or, an LC 2nd order filter arrangement, to match it to a 50 ohm impedance. It may result in a different voltage, and different Ampere level. Nevertheless it should be measurable.
I believe this is similar to the idea mentioned in post #6 ('high Z in, 50 ohm out attenuator network').
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