I'm not sure I follow the logic here. The diff doesn't seem to move around side to side in the body of the car, it's just the back end of your car is all over the shop, particularly on entry and mid corner. A Panhard won't correct oversteer problems.

You say you were on old tyres. That's probably the best place to start. But other things to look for from CARROLL SMITH’S CAUSE AND EFFECT GUIDE:

Corner Entry Oversteer
• Excessively heavy trail braking
• Excessive rearward brake bias
• Severe rearward ride rate/roll resistance imbalance
• Rear roll centre too high
• Diabolical lack of rear download
• Severely limited rear droop travel
• Broken or non-functioning outside rear damper
• Broken or non-functioning front anti-roll bar

Note: A slight feeling of rear ‘tiptoe’ type hunting on corner entry can be due to excessive rear toe-in or excessive rear damper rebound force.

Mid-corner (mid-phase) oversteer
• Driver threw the car at the corner to get through initial understeer – only cure is to educate the driver and/or decrease understeer
• Excessive rear tyre pressure
• Excessive relative rear ride and/or roll stiffness
• Rear suspension bottoming in roll
• Insufficient rear droop travel (non droop limited cars only)
• Very loose rear anti-roll bar linkage

On the topic of Watts or Panhard. Watts is superior. Panhards are biased to cornering in one direction depending on which side the low end is at. You get slightly different traits in left and rights. Watts linkage is better for that, and allows better adjustment of roll centres as well.