It's perspire (rather than sweat) and it depends upon how you define the situation.
I live in Las Vegas, very arid.
We perspire a lot and need to drink a lot of water, but it doesn't show as wet clothes or water droplets on skin.
Glands release water based on skin temperature. At higher humidity, for the same temperature, the water does not evaporate as quickly or easily.
As the air humidity is lower, evaporation is faster.
Now, a body also adjusts over time. I live in a climate always dry and often very hot.
I perspire more naturally now than people who do not live in hot dry climates.
When I travel to a cooler moist climate, I perspire more than the people living there.
But, that is a time related body change.
You are correct, if you define "sweat" as surface water on the skin and clothes not evaporating.
If you define it as the output from sweat glands, the external humidity level should not have a notable effect.