That's partly true.
Several years of drought will definitely weaken the trees.
That's whatr we call "sénescence précoce" in French, prob. something like "early senescence". That's what's happening to a lot of conifers in Europe, esp. in the Jura, smaller mountains near the Alps. The forest can survive a first year of drought, the weaker trees will die, but not so many. In the second year of drought, more, older trees will die, etc.
Sorry, this is in French, but maybe an online translation tool can help :
..
Global warming is by definition global. It can create droughts, or floods depending on the area, or both in a row, which is the worst because heavy rain after drought washes out the soil after a drought.
But maybe it will start to cool down this winter : if more of the ice-cap melts, or if thousands of tons of glaciers detach themselves from greenland, the sea level will rise. At least the people in Florida can keep cool feet...