Using a DB3 diode as a flyback for a 24vdc relay
No, DB3 is not a diode. It is a diac, a non-directional semiconductor switch that can be turned on when its breakover voltage is exceeded.

Breakover voltage of DB3 is 28V. Your unregulated power supply voltage could touch 34V. The DB3 would get triggered and short circuit the power supply, thereby damaging itself and/or the power supply in the process.

A 1N4001 would suffice as a freewheeling diode for this application.