You didn't say if you could ping the second DNS by name ?
If you can ping things by IP address, then you have to assume that IP networking is working fine and the problem is name resolution.
What are the contents of your /etc/resolv.conf file ?
What about if you try:
" and right down near the bottom, it will tell you what the server is that it queried, eg:
;; Query time: 250 msec
;; SERVER: 192.168.144.16#53(192.168.144.16)
;; WHEN: Mon Oct 6 16:17:46 2003
;; MSG SIZE rcvd: 508
(18.104.22.168 is a publicly available DNS server, if it doesn't work, then insert your ISP's DNS server IP instead).