Not so much permanently washed up as looking for a new mission.
The last Templar campaigns in the Holy Land were as recent as 1300-02 [the Templar order was dissolved in 1307], when they captured Ruad Island and were supposed to link up with Mongol forces from Persia in a land campaign. The Mongols showed up in 1300 and took some territory but then withdrew and didn't return. Ruad was lost in 1302 and the Templars retreated to Cyprus. They had been consulting with the Pope on a 'what next' report when the Pope and the French king turned on them.
The Templars' proposals centered around another crusade led by the French and English kings. Neither of them had much enthusiasm for the idea. If the Templar Order had continued to exist, therefore, it would have been back to the drawing board. Which is where this thread came in.