[Dezoris]

Sorry it took so Long to reply.
Sounds to me what you are feeling is that typical delayed reaction fron the front suspension compression to the back kind of catching up with the front.
I would try two things first before adding the sway bar.
I assume your ride height is equal on all 4 wheels?

If so soften up your rear shocks and set the front stiffer.
Take it out drive it hard
Then reverse it.
Take it out frive it hard.
What is better? Report back.

Generally if you stiffen the rear loosen the front slightly increase ride height in the front, you will give the front end more bound, more travel. Longer to compress the front may give you a more neutral feel. also may create more oversteer. Let me know.