Jerry,
From the PIX side you're able to browse with IP computers on the ASA side.
If you manually set the DNS of the computers on the local LAN of the PIX to be the same DNS the local LAN of the ASA, can you PING the computers by name?
In other words, set the DNS of the PIX side to be the DNS used by the ASA internal LAN.
Let us know if this work.
Federico.