How do I keep my home address private following new aircraft registration?
The 'usual' alternative is to have the aircraft owned by a legal entity like an LLC or trust of some kind (trusts are also used by foreigners who own N-reg aircraft). But that only adds a layer of obfuscation: if someone really wants to know who owns the aircraft then they can dig into the LLC or trust ownership details. That information is usually also public although it's at the state level not federal level.

The best thing here is to talk to an attorney in your state, preferably one with aviation experience. He can tell you what legal entities might fit your needs, as well as the pros, cons and costs of each one.