Is this a consolidator or gray market travel agent? Normally Z is indeed upgrade inventory while C is a relatively expensive but not completely full fare business class bucket. My instinct would be to become very suspicious that the travel agent could be using illegally purchased miles. Can you see your exact fare codes or fare rules? Can you see your bookings and receipts (doubtful) on delta.dumb? How do the fares you pay compare to DL published fares for your routes, especially near the time of ticketing?

I'm assuming that these are tickets not using codeshares and written on DL stock, with ticket numbers starting in 006. Look at your booking on delta.dumb using both the PNR or booking code and the ticket number.

Note that for some othe airlines, such as UA and LH, Z is deeply discounted business class, including special sale fares.

I assume that this is not your corporate travel agent, who might well have special contracted fares, but those generally seem to book into normal inventory codes. What do you know about the travel agent and how did you discover them? Have you tried to do some obvious google
searches for problems, including BBB?