I think I'd try and get the puppy in a play pen or start crate training. If they are acting up they are usually tired and enforced naps really make a difference (much easier if they have a confined space they can be placed into and acclimated to the routine of it). A puppy of that age should be asleep the majority of the day (18 to 20 hours)
Tire the puppy out with appropriate play or puppy appropriate chews, put a toy straight back in it's mouth if it bites your skin - keep doing this replacement tactic. Remove yourself from the room for a few moments if there is biting happening that you are unhappy with, rinse and repeat. It's a little young to grasp in totality at this age but the earlier you start the better.
Lots of praise and reward whenever the pup sits quietly, and try to avoid giving attention when it isn't.
If the pup is small enough get a puppy sling and get out and about doing normal things, it needs exposure to the outside world. We had a lot of fun taking ours out in the sling and he often fell asleep in it.
I'd keep the kids busy with whatever I could, focus on setting a routing for the puppy that includes them and not worrying about much else for the week if you can. Eg; teach them appropriate play they can do with the puppy at set times of day, let them have lots of bonding time with it say, in the evening when it's sleepy and not biting so they can form that relationship and interest. The puppy needs to get used to being handled and touched all over its body as much as possible now while it's young is the far easiest time.
Make sure to take it out to toilet every hour or so, every time after any meal or play session and get the DC to carry it outside for that and get involved in saying 'toilet' and waiting for it to go, praising lots and coming back inside.
It isn't easy OP. My puppy is 8 months now and in the teenage stage. I was in tears some days when he was younger and teething/biting) but it does get easier and they are lovely.