With the object Selected on the Artboard- Go to the Appearance Panel (Window> Appearance) simply click on one of the Fills to select only it. It will show with a blue highlight. Then you can click on a new Fill color to assign that color to only that Fill.
Since you mentioned GIMP, here's a simple method
Fill a bacgkround layer with the colour you want
Create a new white layer, and apply the noise filter. Set the layer mode to Multiply
Create a new transparent layer. Choose the Chalk 02 brush, set to a large size, increase the Spacing in the Tool options. Then paint to create a random texture, lower the ...
I think it will be difficult for you to find a filter or effect that will reproduce this in one click. It looks to me like your sample image has several different texture overlays applied to it- one that lightens the base color and one that darkens the base color in places.
I suggest that if you are really into this then try some of your own and come back if ...