There is no footing

It's hard to believe that a home built in the 1950s in Boston would be missing footings. But anything's possible.

I don't like internal water handling systems since they do nothing to prevent continuing water damage, including undermining the foundation. But I suppose that as long as the trenches don't go below the foundation, they may not aggravate whatever support problems have developed.

we have done outside grading and we cannot dig from outside.

I will find out how deep the wall goes down from the floor and make the trench as shallow as possible.

I have opened few feet and found it is not a monolithic structure showing in the bottom link. The wall is NOT sit on top of the floor. The wall goes down few inches and I can see the floor and the wall were separate pour. I hope my understanding is right.

Here is a link that might be useful: monolithic

A monolithic foundation doesn't have to be a slab, as illustrated in this builder's blog where the footings and walls are done in one pour.

At the same time, it would be unusual to have a monolithic foundation enclosing a full-height basement.

If you dig down beside the wall in one section you'll find out for sure.

Here is a link that might be useful: Three Types of Concrete Foundations

same issue with my house. How did you proceed?