Fully disabling Database Cache and Page Cache solves the problem of multiple short URLs (several per post) getting generated automatically in the YOURLS database. (I tried scaling back features of Database and Page Cache before having to disable both entirely.)
It would be nice to make use of all the features of this awesome plugin on sites that use YOURLS: WordPress to Twitter, with YOURLS as the link shortening service. I use W3TC on about a dozen sites, and will continue to use it even if not fully functional.
YOURLS: WordPress to Twitter 1.5.5
Similar issue detailed here: http://wordpress.org/support/topic/plugin-yourls-wordpress-to-twitter-lots-of-duplicates