It is my understanding that the ULN2003A (U4 and U5) has integrated flyback diodes. From the datasheet:

Each consists of seven NPN Darlington pairs that feature high-voltage outputs with common-cathode clamp diodes for switching inductive loads.

Good call on that ground plane. I’ll scoot the relays a little bit to the right. That should resolve the issue.

Thanks for your feedback! :)