In normal english usage, saturated means getting completely wet. It also has a technical meaning, when a liquid contains the absolute maximum amount of a dissolved solid. Saturate can also mean filling something completely, but this is not used in everyday speech. From this comes the business meaning saturate the market - to supply every possible customer or channel, leaving no possiblilities for further sales or development of sales.
Overwhelmed is not an equivalent word, as it suggests that you are not just required to work at 100% capacity, but at more than 100%- the equivalent would be supersaturated. The nearest equivalent would be more like "at the limit of his capacity".
a bit saturated is definitely an incorrect usage, as you can't be a bit absolute maximum.
Likewise, saturated of what I am doing is unusual and unlikely to be appropriate. If it is intended to imply that you are completely busy (and you find it interesting) you would say
I am completely absorbed with what I am doing.
And if you are completely busy and if you are neutral about it, you would says
I am fully occupied with what I am doing.