Sounds like the ZyXEL PK5001Z(Gateway)
IS in bridge mode, and that's why it's not working. It could only provide 1 IP but you had 2 things plugged into it. The reason you could still connect to DHP-306A(adapter) when PC > gateway > adapter is because gateway was still acting as a switch.
Solution: Take Gateway OUT of bridge mode, then it should work with adapter. First do
this step, then do
this one. If that doesn't work I would suggest calling your ISP. According to their site, Bridge mode is ONLY available to Centurylink Business customers. Odd.