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.
I would try Firefly.com with the new text generating option.
I don't intend to deprecate new tools or methods, or dismiss your answer, but... I am concerned that more and more "answers" are "go have an AI system do it for you."
Maybe there's a line between learning a technique and just asking someone to do it for you? Fish vs. Fishing, and all that?
I didn't know about Firefly; I'll give it try too, thanks.
Thank you so much!!! This is so helpful and smart. Illustrator might make more sense to have vector output as you say but I'll experiment in both programs. I did have a distressed typeface in mind (Veneer Three Regular or Nexa Rust), although they make decisions for you about outlines or shadows that you can't always edit). (And sorry it took me a while to reply -- I just saw the notification, not sure why it didn't send.)
Thank you! Ah yes (Thirsty Rough Bol Two is great too!), but you can only see the blue in the type. I'd like to see another color like white specs as well, see. 🙂
Thank you! Nice!