Why spectrum analyzer ruined my oscillator's output?
Your spectrum analyzer has a 50 Ω input impedance, so it is loading the circuit and changing its behavior, unlike the oscilloscope probe which has a very high input impedance (intended to minimize disturbance to a complete, existing circuit no matter where you probe it).

To get consistent results, you should design your oscillator assuming a 50 Ω load, and test it in the presence of that load.

This may require placing an amplifier before the output, to produce a sufficient signal level at that impedance, or it may be only a matching network. An advantage of using an amplifier — called a buffer in this role — is that it also prevents the circuit from being affected by mismatches as much, because the the input to the amplifier is (approximately) unaffected by the situation at its output.

Then you can connect any of


your spectrum analyzer,
a 50 Ω dummy load (non-inductive resistor), optionally with oscilloscope probe in parallel, or
an arbitrary length of 50 Ω coax connected to another device with 50 Ω input


to that 50 Ω output.

This is the conventional approach for modular RF design. The impedance does not have to be 50 Ω but that is conventional for RF test equipment, and it also needs to match the characteristic impedance of the coaxial cable you are using.