"I don't want to be needy"
For many people this is the real obstacle. You'd rather go without than be the sort of person who's always wanting something. You may have known someone like that, a parent or a former partner, and you may have promised yourself you'd never be one.
It's worth being exact about what made that person hard to live with. I doubt it was that they stated what they wanted in a sentence and accepted the answer. People who are felt as needy usually do something else. They want a great deal and want it constantly. They ask in ways that leave no room for refusal, or they don't ask at all and make everyone around them responsible for guessing. They take a no as an injury, and they keep score.
Now look at what the habit produces from the other person's side. They live with moods they can't explain. They're expected to monitor you for signs. When they miss a sign, they're punished with silence. They aren't told what they did wrong, so they can't put it right. They start to feel that whatever they do is likely to be the wrong thing.
That, I'd suggest, is closer to what people mean by needy than a plain request is. The habit doesn't make your needs any smaller. It hands the job of detecting them to somebody else, along with a penalty when they fail. By comparison, a plain ask asks very little of anyone. The other person doesn't have to monitor you or guess or worry. They get a clear piece of information and a free choice.
The same goes for nagging, which people also fear. Nagging is what happens when the need has been signalled and not stated, and is still unmet. It comes out sideways and repeatedly, as complaint. "I suppose I'm doing the bins again." A plain ask, made once, at a sensible moment, is the opposite of that. "Could you take the bins on as your job from now on?" is a proposal, and they can say yes or no to it.
Two things make a request easy to receive, and both are in your hands. The first is that it's specific and has limits. "Could you do bedtime tonight" is easy to say yes to. "I need you to be more supportive" is impossible to say yes to, because nobody knows what it would involve. The second is that a no is allowed. A request the other person can't refuse is really a demand. If they can say, "I can't tonight, I've got this to finish, but I'll do tomorrow and Tuesday," and you can hear that without going cold, then you're simply two adults arranging your lives.
If asking still feels unbearable, it may help to know that the other person will probably be relieved. I can't cite a study for that. My guess is that someone who lives with a partner who goes quiet would give a good deal to be told plainly what's wanted.