Just checked our network folks. No internal firewall that would prevent a connection. Our db server is listening on port 1535. When I use that port in the connection string, I get a connection failure, object doesn't exist. When I omit the port or use a different port, I get the no listener error.
Thanks for your help. Instead of spending so much time on this, I'll try to figure out another solution that doesn't require the web parts.
Thanks again.