maybe if you sketched the block diagram of how you are trying to measure this, we could comment then.
what you described in words does not make any sense at all.
you will either need to find a spectrum analyzer that has these words on the front of it "1 MHz to 400 GHz", OR you are going to need to downconvert your 350 GHz signal with a mixer and some sort of local oscillator. And even if you DID get a setup to work, it would "detect" the AM by showing sidebands on the carrier. You would still need to do more (like look at the envelope of one of the IF stages in the spectrum analyzer)
Do you have any 350 GHz rated diode detectors? They would look like a waveguide input, and a BNC or SMA connector at the output. You could hook a detector output to an oscilloscope and detect AM modulation.