Do exchange rate overshooting models make sense?

Not so much.  Here is the overshooting model for those of you who don’t know it.

So what is the problem?  First, most observed exchange rate movements are unexpected ("news"), rather than forecast in earlier forward rates.  The overshooting model, at best, explains expected movements in exchange rates.

Second, the model relies on a Keynesian money demand function.  Specifically, inflation, operating through a portfolio effect, lowers nominal rates of interest in the initial stage of the mechanism.  Well, sometimes, but don’t count on it.  More generally, the currency vs. interest-bearing assets decision doesn’t have many implications for foreign exchange markets, if any.

Arnold Kling offers some further comments, including this bit:

But in the Dornbusch model, countries differ in terms of their
inflation rates. Inflation is described mathematically as a continuous
movement in prices ("Rudi Dornbusch is the master of the logarithmic
derivative," as Rogoff used to put it.) The swindle, which is present
in all modern macro, is to talk about sticky prices and
continuously-moving prices in the same breath.

Exchange rates are not well understood.  The current best theory is a mix of random walk (but in exchange rates or returns?), noise traders, and possibly some predictable, very long-run, PPP-reverting swings, enabled by the possibility that perhaps traders’ time horizons are too short to compress all of the expected future into the present.  But, unlike what the Dornbusch model predicts, these changes are not well-predicted by nominal interest rate differentials.

Here is a more favorable assessment of the model, from Ken Rogoff.