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It MIGHT work (on the sweatshirt) if the thread used in the embroidery is enough of a different material than the shirt. I've seen t-shirts dyed where the thread was different enough composition that it didn't 'take' very well, and it looked like the shirt was top-stitched in white.
However, I wouldn't risk it if it's important to you. If the shirt was a high percent of cotton AND the thread used in the letters was really shiny, then there's a decent chance that there's enough artificial fibers in the thread that it wouldn't take. But it's still is a risk.
In general, if the attire is very heavy in cotton percentage, you have a better shot of the dye working - though I've never tried dyeing black blacker.
Make sure to run a cycle (or two) in the washing machine after you're done (plus the dryer, too!) to get the dye out!
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