Waterproofing stone foundation

[Guest]

my house, built in the 1930's or 40's, has a sandstone foundation. When there is significant rain, we get puddles in our basement; the water seems to be coming through the walls. We've worked on grading to drain water away from the foundation. The soil around the basement is mostly sand, as we live near a river. There doesn't appear to be drain tile around the foundation. There's also no footing under the walls, as we discovered when we built an addition, so we're afraid to dig very deep for fear of undermining the foundation.

What can we do to dampproof or waterproof the walls to stop the leaking?

[Guest]

I forgot to mention - it looks as though the foundation walls have been parged several times. The area where the leaking occurs does not have parging on it for some reason.

A related issue: areas that are not parged, including the top of the foundation walls, tend to leave a sandy residue over time, which is an annoyance because it makes it impossible to keep the basement clean.

[BasementQuestions]

An interior coating could trap water between substrates and lead to deterioration of foundation.

Possible solution with having questionable footer for foundation, would be an on the floor system with a vapor/moisture barrier that tucks into the system.