Adjustments made before during a game can win...or lose a game. Pre-game scouting reports or prior encounters can give clues for how each team's offense and defense should come out of the gate. However, there is no substitute for actually playing the game and making real-time evaluations. There is a fine line to walk between jumping the gun on an adjustment and not acting soon enough and I think that one of the hardest jobs of a team strategist is this evaluation and subsequent decision for adjustment or non-adjustment.
In the given situation, it sounds like the defense tried to take away the opponents first option (banking on a low likelihood of being beat on the opponent's second or subsequent option), but the opponent made an effective pre-game adjustment, and kudos to them. Halftime is a natural time for adjustments to be made to happen and in this particular situation, I would absolutely change up the defensive plan.
In close games, teams trade points, teams take turns going on runs, and at the most basic level, teams settle into a rhythm. One of the goals of a defense should be to make the offense uncomfortable, and clearly the opposing team is feeling pretty at ease with their star thrower being forced away from the disc. The decision for whether the defensive adjustment is a zone, clam, a straight up mark, backing man defense, etc., should be made with your team's strengths (and other external factors, like weather) in consideration. If the decision is to stay with man defense, the other factor to consider is the type of defender to put on the thrower. For example, on other team's big throwers, there can be much success to be had with rotating smaller-in-stature handler defenders who specialize in challenging the dump pass with more physical defenders who cover and hold lots of ground on D and provide a huge mark.
Whichever method ends up being chosen, the goal with the defensive adjustement should be to disrupt the offense's rhythm enough to get a couple turns (and defensive scores) and shift the momentum of the game.