It is good to distract with toys where possible but as you say not always possible!
Another thing to try is to hold food in your palm under your thumb so he can smell the food but has to work a bit to get it. As he is trying to get the food gently and slowing stroke him. Long slow strokes down his back. This gets him used to being stroked and less likely to bite initially when he sees a hand coming at him.
Also teach him a hand touch. Just put a flat palm near his nose and he will move towards you hand, say a marker word eg yes, and put a treat into your hand to let him eat. This helps to give a new meaning to the value of hands.
As he is a spaniel I would be using all of his food/meal to train BUT do not get tied up on formal training. Throw a bit of his food a short way away from him, he runs to get it and will then turn to you, mark the turn say his name and then he will come to you and you treat him.
So basically you are throwing some of his food out and then feeding when he comes back to you - this is the beginning of recall. It also helps in that it will tire him out and give you a bit more peace.
As he gets used to your hands meaning food you will then be able to lure him into sit positions, middle positions etc and do more formal training if you wish.
A very long toy is good to stop him biting you feet and legs as you move.
Re carrying to take him outside can you carry him like a "log" so rather than have him upright and face near your face can you carry him horizontally across your tummy? He is then more likely to bite your clothes(!) than your face. He may find the environment more interesting when out and that also may help you. But do not go for long at all 5 mins is perfect.
What tends to happy with a bity puppy is that people approach the puppies either tentatively or tend to grab them to avoid being bitten (tbh all very senseible precautions!) however that can also be more exciting and make the puppy more jumpy and then increase the biting and the circle continues.
However saying that you have done nothing to cause this some dogs are just bitey buggers!