Yeah, shimming the cores is the second best way to airgap it; grinding the center peg is the best way. Which can be done with a rotary tool and grinding wheel, but it's a bit messy and dirty. Ferrite is brittle, so go carefully. You have to grind away a bit, check inductance, and back and forth until it's the required value.
There are various ways to measure SRF. The most basic is to use the component in a series impedance divider. Adjust frequency until the phase shift goes away: at this point the component has the same phase as the other element. In a divider with a resistor, this means the component is resistive, which means it's at resonance (L and C cancel out).
So:
Signal generator from Vi to GND
Component from Vi to Vo
Resistor from Vo to GND
Scope probe from Vo to GND
Use a 10x probe, and use a smallish value resistor (a few k max). This minimizes loading from the probe (which is capacitive in the ~MHz range).
It could also be tested with a grid-dip method, using a coupling link (i.e., a small coil) and whatever coupling gets through the ferrite core (pot cores are well shielded, but not perfect; or a small loop could be put into the winding window or near the air gap to get more coupling). In this case you're testing for the frequency where the link/loop/antenna sees a dip (notch) in amplitude response. A sweep generator and oscilloscope (set to peak-detect mode) is one way to do this. (This isn't as easy a method overall, so I'm not bothering to get very specific; it used to be more common, with the "grid dip meter" being fairly common equipment. Which would be easy enough to make, it's true, but not so easy to calibrate, is the problem.)
As for measuring inductor parameters with limited equipment, I use these for various things:
https://www.seventransistorlabs.com/Calc/RLC.htmlTim