Find your nearest monumental masons or high end kitchen fitters. Both should have equipment to cut/polish stone. Get them to cut and polish some half inch slices, see if you've got some nice grain/growth rings preserved.
Can I cut them with a tile saw or will that scar them?
@azzow2 a tile saw should be ok, but watch out - petrified wood from the southern US is usually amorphous silica - like flint, chert, agate etc. so really hard stuff. The cleaner you can cut, the easier the grinding/polishing will be. That's why I'd say take it to a mason, because they'll have better equipment.
As you probably are aware Arrow living in NM as we do, its not uncommon to find petrified wood all over New Mexico and Arizona. As the Mesozoic seas retreated on either side of the Rocky Mountain complex, large swaths of shallow marshes persisted as the inland ocean retreated.