Turntable feedback rumble problems with new sub

Unfortunately the table and speakers sit on a big media cabinet from Design Within Reach which is not ideal.

Sounds like the media cabinet isn't doing its job. Can you hear foot falls when you walk past the turntable while its playing?

Other than placing the Table in another room its a hit and miss
scenario.
I assume your not using the table with the dustcover down
while playing or have the sub level and crossover adjusted 
to high?
As noted there are lots of tweaks but what many people 
continue to look the other way from is adequate room treatment
which will ameliorate room modes.

All you need is a couple inches of sand in a box. Pack the sand, put something on top of the sand- can be one big sheet of MDF or four small ones, hardly matters- and put the turntable on that. The footers you have may work better under the sand box, or under the turntable, you will just have to try them to see.  

If you have the space and the funds you can do even better with a granite machinists plate, which is what my table is on. Between the granite and the concrete rack is an inch of sand. On the bottom shelf the phono stage is on sand and if you look close you will see the square MDF footers the phono stage is on. Pic #7. https://systems.audiogon.com/systems/8367
The sand is mixed with a small amount of mineral oil that helps it pack smooth and stay put without dust. Dirt cheap, tried and true.