First of all, for similar phrases like this, I never like to say that they are interchangeable, because that implies that they connotate the same thing. There is usually a best choice to make. While the denotation of the phrases might be interchangeable, providing an incorrect connotation could lead to confusion on the part of the reader or listener of your sentence.
So let discuss connotation with an example.
In doing so:
Bob walked his dog. In doing so, he exercised.
The first phrase in doing so gives the impression that effect of his first action (walking the dog) wasn't at all intended to cause the second. He exercised, but this occurred only as a side benefit.
In this way:
Kathy finished watering her plants and set down the hose. In this way, the hose would be close by for next time.
In this example, the phrase in this way is used. It is implied that Kathy desired the results described in the second sentence. She wants the hose close by. However, it isn't a very strong desire. If her husband comes and stores the hose, she will just get it out again.
So that:
Fred went to Law school so that he could get a degree.
In this example, Fred performed the first action (law school) expressly for the purpose of achieving the second action (get a degree). This was his entire goal.
I'll change out the last example with a different phrase for clarity:
Fred went to Law school. In doing so, he could get a degree.
If you said this, a listener might be confused. They might ask "Did he want the degree?" or "Did he have another reason for going?" Maybe Fred's main purpose was to party with college students and blow off classes.
Conclusion:
Remember that changing out these phrases for each other will still make sense. It will still be grammatical. But it will connotate a different mindset on the part of the subject.
It means being extremely eager to do something, especially when the haste to do it right away causes problems or errors.
When the Admiral sneezed, junior officers were falling all over each other to pass her a tissue. Two even collided with each other and had to go to sick bay.
Imagine a group of navy officers all pushing one another out of the way to get in the Admiral's good graces.
When Boeing announced their intent to open a new plant, State Governments were falling over each other to offer tax incentives and favourable legal environments.
State representatives, in their eagerness to get a big contract for their state, might find themselves offering so many incentives to the company that they end up losing money overall.
When Michelle, my high-school crush, asked for help with her homework, I was falling over myself to do it for her.
So eager to help the girl that the speaker perhaps put themselves in a bad position.
Best Answer
Neither is incorrect, although it sounds far more idiomatic without "in", and that isn't a personal preference.
"Way", in this context, is often used interchangeably with 'style' or 'manner', both of which do require "in", for example:
However, "way" literally means a route, or a direction, not strictly a style, and is more synonymous with 'method'.
Saying "in this way" can change the meaning - for example, if we said "walk this way", it would idiomatically mean in this direction; however, if we said "walk in this way", it would mean in this style, or manner.