Anytime I read a post about a new dog owner even thinking about giving up their puppy, it makes me worried, cringe, and feel sad. Please don't give up on him. You brought him into your lives, you need to try harder to get past this stage. Forget about your embarrassing situation at the groomers for a second...
It's really important to understand that what you're experiencing is puppy behaviour. It doesn't mean it should be tolerated, but as the puppy owner, you need to be in control to train him out of bad behaviours.
As others mentioned, Yorkie puppies (until ~1.5 to 2 years) will have an immense amount of energy. They will try what they want to do until you train them not to. Slapping them doesn't really work - nothing physical will work. Repetitive No's don't work either - they will learn to zone you out. Locking him in the crate won't work either - you're just teaching him to hate the crate.
What works best is positive reinforcement and a positive mindset. Don't ever let your frustration get the best of you. They can sense it and feed off of it. What you need to train you dog is impulse control.
All puppies want is food and your attention. That's it. Start with obedience training, either on your own or take a dog training course. I'm a huge fan of the late Dr. Sophia Yin. I learned from her that dogs should sit to say please. It's a really good start for impulse control and have a calm dog. Even puppies can achieve calmness
Say Please Protocol (or “Using Real Life Rewards”) | Grisha Stewart https://drsophiayin.com/videos/entry...se_by_sitting/ https://drsophiayin.com/videos/entry...ddenly_settle/
Once your puppy learns some obedience, everything else is easier to train. Spend 10 minutes with your puppy multiple times a day for training. It's a LOT of work. But it REALLY pays off. Honestly.
There are two ways to expend their energy levels: mentally by daily training and physically by nice walks outside. I'd suggest to do both!
For the biting. Make a LOUD YELP noise, even if it doesn't hurt. Then, turn your back and ignore him. Only when he doesn't bit you, does he get any of your attention. This works for barking at you too. If you react to him when he bites/barks, then you're reinforcing him to continue these behaviours. If the yelp noises don't work, use your hand/arm to push him gently away and then turn/walk away. Whatever you do, prevent situations where your hand is near his face.
Make sure he has a large variety of chew toys.
Anyway. Start with this. Just don't give up so easily