The breaker serves to protect the wiring, not the device(s). There is no problem plugging a 1A device into a 15A or 20A (or 30A) circuit.
Spork's idea is fine, safe, and legal. I'd approach is as:
1. Turn off the 30A double breaker in the panel.
2. Remove the Nema 10-30R behind the dryer and see what you have for wiring. It will likely be a 10-3 (black, red, white, and bare copper). If it doesn't have a bare copper for whatever reason, you're going to have to either pull one (surely legal) or use the red wire marked-as-green (not to code, because of the size of the wire doesn't permit re-marking, but is electrically safe as long as no one later misunderstands its purpose).
3. Wire the new dryer outlet - black to brass, white to silver, bare copper to green, and wire nut off the red as unused. I'd buy a hospital-grade 15/20A circuit (has a T shaped connector). They run about $10 instead of $0.50. Don't back-stab the connections; use the screws.
4. Turn the breaker back on and check your work.
5. Optionally (but I'd do it), replace the 30A double breaker with a 20A single breaker (turning the power off/on as is obvious, though I'm comfortable replacing a breaker with the main breaker still turned on so as to not inconvenience the whole house). Install a block-off plate to cover the hole you left. Label the red lead and rather than wire nutting this end off, I'd tie it into the neutral bus bar.