There are always new questions, these continue until one realises one is helping others. This is when life gets more interesting.
I omitted making any mention of current, Common Mode, chokes (aka baluns) at the feedpoint, Raj had put that forward, and so has Iz..
Without such a choke at the feedpoint the outside of the feeder coax will ALSO be part of the radiating aerial, and could be making part of an extra resonance. Certainly without such a choke you will easily get the 'RF in the shack' which you now complain of.
My solution is to this phenomenon would be to make a balun/current choke for the feedpoint, and another one for the feeder just before it enters your house or shack. These should stop the RF energy that is obviously following the cable down from the antenna, they won't stop direct radiation from the antenna.
Another factor in clearing RF feedback is to ensure you do have a resonant load in the antenna, looking at your measurement results you appear to have a very low Q antenna, I doubt if you are radiating much energy at all.
Go back to Start! - Remake your antenna with a current choke in the feedpoint. Make sure you trim the 40m section first, and then the 20m part.
Using a ferrite core in the current choke will allow it to cover a wider bandwidth, but at a pinch one can wind feeder cable around any magnetic material, even a mild steel bolt will make a difference. Although the most effective, Ferrite Rings are not the only
sort of possible core material.
One expects to see on 40m something like a 1.4:1 to 1.7:1 response at 7.1MHz rising to somewhere over 2:1 or 2.5:1 at the band edges. These are not catastrophic results, instead they are expected.
On 20m one should expect higher values at the band edges maybe 3:1 to 3.5.:1. In Region 1 the band is almost twice the the span of 40m.
As another has pointed out 1.1:1 or even 1:1 matches are not the be-all or end-all aims.
Off you go and experiment a lot more.
And just maybe you'll find time to make contacts on the air, I look forward to a QSO, on 20m?.
Remember to waterproof all the coax connections with vulcanising tape and then cover with black electrical tape to prevent degradation in sunlight.