Looking for a database that supports fulltext but does not lock table on insert or update
With the current MySQL version this is not possible. Apparently the next version (5.6) will support full text indexing on InnoDB tables. I have no idea when that version will officially be released.

You might also want to consider an upgrade to PostgreSQL which has pretty decent full text indexing (I haven't really used that, but it appears it has some additional features like stemming that MySQL does not have). 

And PostgreSQL copes very well with concurrent reads and writes.