What's commonly used to seal cracks in concrete foundations where I live is called "hydraulic cement". This is a cement mixture that expands as it cures, and that causes it to form a tight seal in the crack.
PS:
I'm going to start out by saying that I'm certainly no concrete or foundation specialist, and there are people in here that know much more than me about both. However, I just don't see why water getting into a crack between a sidewalk and a concrete house slab would cause either one to lift.
Still, I don't understand why that crack would cause anything to lift. Water that gets into the ground will continue to sink into the ground until it gets down to the water table. If that water were to freeze and expand, it could lift the sidewalk. If you have a lot of clay in your soil, the clay would expand and lift the sidewalk, but one would expect that the clay under your sidewalk would already be wet since any moisture in it couldn't evaporate easily, and so adding water wouldn't result in any further swelling of the clay in the soil. And, I see you live in Texas so the ground isn't going to freeze deep enough in winter to cause any significant amount of frost heave to lift either the sidewalk or the house.
I'm wondering if the smarter thing to do wouldn't be to just move the sidewalk away from the slab, or cut a few inches off the edge of the sidewalk and fill the gap with stones so that the sidewalk could move up and down independant of the slab, and just let it do that.
I guess I just don't see any reason why water leaking in between a concrete sidewalk and a concrete house slab would cause either one to lift unless that water freezes. Did you're foundation specialist explain how that could happen?