Yes this type of insulation will settle and also hold moisture.
In fact, almost all insulation will hold moisture. However some insulation if allowed to be properly ventilated will dry out and not allow growth of mold or mildew.
Also wet insulation does not perform at the manufacturers R value specification and must be removed and replaced.
As to the sheet rock and stucco pulling away from the wall at the windows, this is a clear sign of moisture infiltration and swelling of the wood structure and sheathing.
Properly installed stucco MUST be a water managed system especially if it is installed over wood framing. A water managed system allows for any moisture which may get below the stucco to drain or evaporate out. This system incorporates multiple layors of vapor barriers and termination strips with weep hole perforations.
One telltale sign if this is the type of system installation is the presence of termination strips at the base of walls and above windows with perforations to allow the moisture to weep out. If you do not see this type of termination strip your stucco will not allow for moisture penetration and could lead to internal wall rot.
Test for leakage around the windows. ASTM has protocols in place for water test for windows which can be performed in the field. This type of test incorporates high pressure water jets mounted on the exterior of the home and the use of a thermal imaging camera on the inside of the home. Which the water is being sprayed on the exterior, measure the temperature around the window. If a leak is present you will notice and change in color to dark blue at the location of the leaks. In some cases it is recommended to remove a portion of the sheet rock for a better view of the interior of the wall.