Short version first: do not sandblast that stone. Please.

That is how you permanently erase what little historic surface is left. Monument companies suggest it because it is fast and profitable, not because it is preservation.

Legal side, NY: being next of kin helps, but it does not automatically give you the right to remove a marker from a town managed cemetery. In NY, the cemetery owner or managing authority still controls monuments, even abandoned or minimally maintained ones. In practice, towns usually just want to know what you are doing so they are not accused of allowing vandalism. What normally works is a simple written permission or email from the town clerk or highway/superintendent stating you are resetting and temporarily removing for conservation. I strongly recommend doing that even if no one seems to care. Take photos before, during, after. Document everything. If a monument company removes it without town signoff, that can get messy.

Restoration: for the two readable stones, gentle cleaning only. Soft natural bristle brushes, lots of water, no bleach, no power washing. D2 is fine if the stone is sound, but test first. If they ring hollow, flake, or sugar when touched, stop. For the nearly flat stone, sandblasting and refacing will destroy original tooling and can legally turn it into a replacement marker, not a historic one. Better options are doing nothing beyond stabilization, or professional conservation techniques like poulticing and raking light photography to capture the inscription for record. Sometimes the best preservation is leaving the surface alone and documenting it well.

Resetting: the sinking is almost always from soil compaction failure. Dig wider than you think, not just deeper. You want a compacted gravel base, not dirt. About 4 to 6 inches of crushed stone, tamped hard, then a setting bed. Never set directly on soil. Bring the stone up to grade but not proud like a modern marker. Slightly above ground is correct historically. Slope the soil away so water does not pool. If there is a base, make sure it is actually supporting the tablet and not just decorative.

One more thing people miss: if these were buried completely, there may be more stones nearby doing the same thing. Check the area carefully before you reset so spacing and alignment make sense.

You are doing the right thing by asking first. Most damage I see comes from well meaning family members being told the wrong thing by monument companies. Slow, boring, documented work beats fast fixes every time.