(Edit to update):
Here is how I would route your board:
relay_boards_03.fzz (56.1 KB)
and the what and why to get from your original to this:
1) Load new parts Header_4_fixed.fzpz and AZ832P2-2C5DE_Latching_relay_only_active_pins.fzpz
Header_4_fixed.fzpz (5.4 KB)
AZ832P2-2C-5DE_Latching_relay_only_active_pins.fzpz (8.8 KB)
2) Select the relays one at a time and in inspector change varient from "varient 2" to "only active pins". This deletes the unused pins in pcb and makes drc happy and routing easier.
3) Select the connectors one at a time and change varient from "polar lock" to "fixed PCB footprint" to correct the connector spacing.
4) Add bypass capacitors (2 .1 uf ceramics and a 10uf tantalum) to the power supply.
5) Separate logic power and ground from relay power and ground. The relays are likely to cause current spikes and induce noise in to the digital ground if not separated.
6) move everything possible to the bottom layer (makes trouble shooting easier in case of error!).
7) Increase the trace width on power and relay connections (both likely to be high current) to maximum size. The 74HC595 probably doesn't matter, but it is easy to do and won't hurt anything.