Your reasoning is a bit flawed.
The immediate effect of melting of the ice sheet is to add a small amount of water to the oceans (here small is relative to the huge amount of water already in the oceans). Over time, this small effect will result in sea level rise, currently estimated to be a rise of 50 to 200 centimeters by the end of this century.
At any one spot over the ocean, the rate at which water evaporates is a function of
- The surface temperature of the water,
- The surface temperature of the air,
- The humidity content of the air, and
- Surface winds.
Evaporation rate increases with increased air and water surface temperature, decreases as the relative humidity increases. Very calm air results in a thin layer of fully saturated air just above the water. This thin layer diffuses into the atmosphere as a whole. Diffusion is a slow process. A slight wind significantly increases transport from that surface layer. An even stronger wind keeps that surface layer from forming.
Integrating this local evaporation rate over all the waters of the Earth results in a global evaporation rate. Global warming on average increases both air and water temperatures, thereby increasing the evaporation rate.
Melting of the ice sheets does result in sea level rise, which in turn increases the surface area of the oceans, which in turn increases the global evaporation rate. This is a rather small effect compared to that of global warming. The sea level rise at any point in time is rather small. It only becomes significant on multi-decadal time scales. Moreover, even a two meter rise in sea level won't increase the surface area of the oceans by that much. Melting of all of the ice over Greenland and Antarctica most certainly would, but that won't be a problem for many hundreds of years. The immediate problem is what happens over the next 85 years or so, to the end of this century.