Parasitic Capacitor current of an inductor
Yes your characteristic impedance is too low (L/C depenent and coax) ,
meaning the resonant frequency is too low,
meaning the interwinding capacitance is too high. 

Choose a design where the L/C ratio with lower C meaning higher Zo and LC product is reduced to give higher fo, and L/R ratio gives lower Q.

suggestions.

This may depend more on your coaxial load or choice of dielectric as well as inter-winding space.

Air-gap clamshell, bifilar wound gives least L but higher C unless larger radius semi-rigid.

It is easy to measure Q and SRF. Do you know how by measuring current with sweep?

Try to make pulse look like my air coil current sense.


Note

area of -ve pulse matches +ve, but rise time limited by scope..