Can't get Mac to use correct version of sqlite3

I know variations of this question have been asked before and I've tried the recommendation listed in the other postings. I'm trying to get sqlite 3.11 installed in El Capitan using brew.

I did...

$ brew install sqlite3
$ brew link --force sqlite3

The --force was necessary because I got an error without it indicating Warning: sqlite is keg-only and must be linked with --force. Also, I have /usr/local/bin at the front of my PATH. So everything seems like it is set up correctly. Now I do this...

~ $ which sqlite3
/usr/local/bin/sqlite3
~ $ sqlite3 --version
3.8.4.3 2014-04-03 16:53:12 a611fa96c4a848614efe899130359c9f6fb889c3
~ $ /usr/local/bin/sqlite3 --version
3.11.0 2016-02-15 17:29:24 3d862f207e3adc00f78066799ac5a8c282430a5f
~ $ ls -l /usr/local/bin/sqlite3
lrwxr-xr-x  1 me  admin  35 Feb 16 19:12 /usr/local/bin/sqlite3 -> ../Cellar/sqlite/3.11.0/bin/sqlite3
~ $ /usr/local/Cellar/sqlite/3.11.0/bin/sqlite3 --version 
3.11.0 2016-02-15 17:29:24 3d862f207e3adc00f78066799ac5a8c282430a5f

I'm at a bit of a loss here. It says it's using the correct sqlite3, but the version is wrong. When I look at the homebrew version of sqlite3 that is linked to the one in /usr/local/bin it has the correct version number. Any thoughts on what I'm doing wrong here or suggestions on how to remedy the problem?