Here are my top go-to techniques to add effects like texture and dimension to screen printed design.
One technique starts with drawing your own shapes. Use any shape you like. Once you choose your shapes, colorize them in different percentages of the same color to keep the number of colors down while still giving the illusion of multi-color design. Then, paste these shapes inside of your text by creating a clipping mask.
You also can use graduations to create interest, but be sure to create them using spot colors. This gives the illusion you are using more colors when you are actually blending two colors together to create a third.
3. Drop Shadows
Create a quick drop shadow to add dimension by duplicating your text layer then coloring the bottom layer black (or whatever color you want the shadow to be), then offsetting it slightly.
4. Distressed Bitmap Overlay
Another option to create texture is to add a distressed bitmap overlay to create a worn effect. This look is popular on shirts.