I was the exact same way and baby came the next day. I'd continue to labor at home but call if contractions are closer together. Mine didn't feel close at all and were very mild but I suspected my water had broken so I went in and was already 5cm and fully effaced 😂
I would call maternity triage and let them know. They likely don't want you in straight away, but will tell you when you should come
And yes I reckon partner needs to stay off work!
It might be worth calling them and just asking to check if your waters have gone as well x
I would say go to L&D it sounds like you close to having your LO soon