How do I bring up a raise without sounding demanding?

Frame it as a business conversation, not a personal ask. Lead with specific contributions and results, then name your number. "Based on X, Y, Z, I'd like to discuss moving my salary to $X" lands very differently from "I think I deserve a raise."

What if my manager says the budget isn't there right now?

Ask what conditions would need to be true for a raise to happen, and by when. You want to leave with a concrete timeline or criteria — not just "we'll see." That converts a no into a roadmap.

Should I mention a competing offer when asking for a raise?

Only if you actually have one and are genuinely prepared to act on it. An empty threat is worse than none — experienced managers test it. If you have a real offer, present it as information, not an ultimatum.