“rainfall correlates negatively with temperature”? I don’t know why you would think that: Antarctica is the driest place on the planet, whereas you have monsoons at the equator.
A warmer world means a higher vapour pressure, so higher evaporation, and therefore more clouds and more rain: This is not my opinion, it’s a fact.
The research you provided a link to is completely irrelevant to the above, because it does not consider sea temperature (higher = more cloud formation) but land temperature. Logically the rain falls from the sky after a cooling of the cloud to the dew point normally over cooler land, so we are talking about two separate effects: Weather vs Climate.
However the shrinking of deserts over the last 25 years is not attributed to the warming of the Planet (negligible so far anyway) but to direct CO2 fertilisation affect. Here’s the CSIRO research results proving that:
Deserts 'greening' from rising CO2
Increased levels of carbon dioxide (CO2) have helped boost green foliage across the world's arid regions over the past…
Basically, the climate effects of CO2 have been exaggerated, while the benefits of CO2 have been ignored by the IPCC.
Here’s more reading for you: I hope this helps!