-26 or 27 ft. (as far as I know, the max. length that still qualifies as trailerable)
-outboard engine (avoid hassles of inboard)
Oh yeah, max. budget 20K or so.
If trailability is a factor then you need to look at width, not length. I believe that Catalina 27 is wider than max legal width without permits (102 in)
For 20 grand there are virtually hundreds boats in 30 ft range, perfectly suitable for live aboard.
However, going back to your requirements
Bristol 27 with outboard motor well and “dinette“ interior can be found for $6000 at most.