I agree with the others who said you probably have an acoustic feedback problem. Acoustic feedback is a mechanical vibration where sound from the speakers travels through the air or furniture and enters the turntable's tonearm, platter, or pick-up. Once in the pick-up it gets amplified and sent through the speakers again, and again.
The best way to deal with acoustic feedback, as TLS Guy mentioned above, is to move the turntable or the speakers, until you no longer get the noise. You already did that when you moved your new Teac turntable to the top of the cabinet and the severe hum disappeared.
You may not have a choice. As a test, try putting some material (newspapers, magazines, folded towels, etc.) between the sliding shelf and the turntable to dampen vibrations. If that makes a difference, you can keep that or look for a more permanent dampening material.
Avoid things like screws or a granite slab, that would couple the shelf more securely to the cabinet. Look for vibration dampening materials such as a rubber pad. No need for it to be expensive.
By "way too tight" do you mean too short? 4-5 feet long is alright, but don't go longer. If there is enough slack in the cable, you probably don't need 90 degree plugs on one end. If you really need them, I don't see a reason why something like these couldn't be used for a phono connection.
Amazon.com: Cable Matters 2-Pack Right Angle RCA Adapter, 90 Degree RCA Adapter: Home Audio & Theater
www.amazon.com
Does the back panel of your turntable look like the photo below? It apparently has an internal phono pre-amp and a switch (labeled PHONO EQ) to allow you to use it (ON) or bypass it (THRU).
TLS Guy suggests you might also have induced hum from your integrated amp. If so, slide the PHONO EQ switch on the Teac to ON. You must also move the cable connection on the back of your integrated amp from the phono input RCA jack to any other unused RCA input. That will raise the signal in the cable between the turntable and the amp to a higher voltage that may make it less susceptible to hum.
If you use the McIntosh's built-in phono pre-amp, be sure the turntable is switched to THRU.
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