adding three colors to a distressed typeface?

I think you can do this as live text entirely in InDesign. See https://creativepro.com/creating-see-through-text/

Hi @Katherine25238449muqx , Your screen capture looks like a simple drop shadow. Here I’m using Adobe Fonts’ Thirsty Rough Bol Two:

I would use Photoshop, with some adjustment layers, for specific-sized use, but Illustrator would produce scalable vector output that might be more suitable — more usable as fonts are, rather than images.

Go with Illo. Place your text at a fairly large size, convert to outlines, modify as you like (using layers helps), then save and import to ID as the AI file directly. You'll have to do each block of text as a separate file, but you can export from individual artboards to keep all the work in one file.

ETA: I assumed you were starting with a distressed typeface that would have elements to work with. If you want to to start with a regular typeface, you'll have to apply a texture to at least the area inside an outline or embossed edge. That might be easier in Photoshop; you can probably find a premade texture that can be used or 'massaged' into the look you want.