Installing 3 Gang Smart Switch
You can't do that
The dining room, and kitchen, circuits are old-school switch loops.  They only bring always-hot and switched-hot to the box.
You are poaching neutral from a different cable potentially on a different circuit or even different phase.  Can't do it.
You can't attach the smart-switch wires to anything except wires in that same cable.  All related wires must be in the same cable or conduit, NEC 300.3.  There are several reasons, but one relates to the current imbalance it causes, resulting in EMFs not canceling each other out as they normally would.
That means you will need to either go with a different smart switch, or use a smart module up at the lamp itself if power enters there.
As far as the blinking problem, it might be tangled up in this, but it might also be a problem of the light or bulb not being dimmer-ready. Try swapping an old-school incandescent in there and see if that fixes it; if so it's the bulb.