Does this checkers-like game exist
This strikes me as a modified two player version of Peg Solitaire, rather than checkers.

Notably:


Players alternate turns except that a player is granted additional consecutive moves (i.e., captures) during a single turn, under certain circumstances.
Players must readjust two separate pegs prior to their first move of a turn, then they must move using a third peg. Consecutive moves must be made with that same peg.


This, of course, changes the win condition from "remove all the pegs, but one" to "mess up your opponents peg-capturing board state, worse than they mess up yours, and whoever gets the most captures, wins"

Best of luck marketing this game as well as the new Pandemic-killer, "fully cooperative peg solitaire," coming soon to your dreams!