I will answer your liveaboard question. 23 foot is going to be hard to live on with the requirements that you mention. There are some tubby 23 footers with a lot of amenities but they are boring to sail. They are heavy unresponsive boats meant for long passage making. You would quickly get bored of sailing if you bought one of those as your first boat. They are also pretty rare and expensive for what you get. The Dana 24 is a perfect example of this type of boat although there are several others have been built over the years.
Now if you move up to a 28-30 foot you are going to start having the typical livaboard amenities. You will get a head, maybe a shower, enough storage for all your possessions, provided you don't have a lot in the way of possessions. There will even be room to have a water maker. Although I water makers are expensive and use a lot of electricity.
Hunters, Catalina's and Beneteau are mass production builders that have been building 28-30 foot boats like these for years. Although in the 70-80s and later there will be many boats that fit into this category. Irwin, Sabre, Oday to name just a few.
The $6000 price limit is going to be tough on a decent boat of this size. Maybe you can cram yourself into a decent 25 footer. No shower, very limited storage, probably a head. If you step up to 10-15000 you can get into a 28-30 foot boat. I agree with some of the prior posters that 23 foot is a nice size to learn on but I would hate to live on it.