Great. Thanks for the additional information. I really appreciate it.
So the first thing I'd do would be to stop playing with the tug toy. The reason being is that you're actually teaching her to use her mouth to play with humans...it can actually send the wrong message to her.
Mouthing is a very, very common behavior in puppies and young dogs. They have deciduous (baby) teeth that are becoming painful and needing to come out to make room for their grown-up teeth (just like in human children who lose teeth). Every breed has different times when this teething period happens, and in some breeds, it can last upwards of a year! Every breed goes through this period, but herding and/or hunting breeds have been bred to move or retrieve animals with their mouths, so they're just naturally more mouthy and will continue this phase much longer than some other breeds. Additionally, young dogs who weren't taught to not mouth while in this period will continue this behavior even once they're adults...simply because they don't know any better. This could be what is going on with your dog.
As I'm sure you know, Poodles were originally bred to be water retrievers, so it's not surprising that she is using her mouth on your when she wants something.
Regardless of what breed of dog it is, being bitten/mouthed is painful. Deterrents such as Bitter Orange or Bitter Apple (found at most pet stores and your vet's office) can be effective because it tastes awful. Most dogs won't continue to put your hand in their mouths once they get a taste. Of course, nothing is 100% and there are exceptions to every rule (I have a Husky who doesn't seem to mind the taste and will actually lick things that have Bitter Apple on them). You could also try dabbing your hands with dill pickle juice. The juice contains an ingredient called alum..it's used to keep the pickles fresh tasting, but is very bitter. You could try this option if you don't mind smelling like a giant pickle.
That being said, it's actually a better plan to train your dog not to mouth you instead of relying on deterrents. Your dog is bonding with you, and she needs to learn that this behavior hurts you. One of my favorite suggestions at this point is to get her into a basic obedience class. Not because you want her to learn to sit and stay (although that's pretty great too), but because this will help her view you as the top dog in the house and as top dog, it would be disrespectful for her to put her mouth on you.
It's also good to remember that puppies are taught bite inhibition within the litter. When one puppy bites another too roughly, his litter mate will yelp. This teaches the biter to soften the play bite. So something you could try would be when your dog uses her mouth on you, you should yelp loudly (or use a stern "NO!"), stand up, and walk away from her. This will send the message to the dog that what she's doing isn't okay with you and when she does it, it ends playtime. Your dog is going to want to do things with you, so by walking away when she puts her mouth on you, she'll figure out that mouthing you ends the fun.
That being said, it's definitely possible that your dog is simply at an age where she really NEEDS to chew on something. So when she gets mouthy, substitute something appropriate for her to chew on. Rawhides,rope toys, or Kongs (indestructible chew toys that you can fill with peanut butter) or any other chewy toy are all perfect alternatives to chewing on the humans.
I'm sure you're not doing this, but please make sure you never smack her muzzle or hold her mouth shut as a way of disciplining her. These behaviors can actually cause a dog to bite more, and harder!
I hope this helps!!