I asked a question recently which was closed as being too broad. I disagreed with the reasoning, but I understand the position that programmers.SE moderators took on it and I want to ask questions that are less opinion-based. My question is now... if I delete that question, will that remove my question block so I can ask a different question? Or do I need to simply wait it out? Not trying to game the system here, just wondering if deleting it is enough of a "good deed" to terminate the question ban.
Since this applies to all StackExchange sites, it's covered on Meta StackExchange in this FAQ Q&A. This is the part that answers your question:
Before you do anything else, fix your existing posts! As noted above, down-votes cast by the rest of the community factor into the ban - so the single best thing you can do to get it lifted is to address any objections raised by others. Were your past questions unclear? Did they fail to show any effort on your part? Poorly worded, titled, formatted, and overly long or short? Then fix them!
Note the emphasis on fixing. Do not delete your posts. As explained above, deleted questions (if less than 30 days old when deleted) do still count towards the question ban. Deleting your posts does not help to lift the ban. Only fixing does! Under some conditions you can see a list of your own deleted questions and answers.
If you are banned from asking questions, then writing a few quality answers that get upvoted might enable you to ask questions again. But as the internals of the filter are secret, there is no way to know for sure.