Is it possible to be right exactly adjacent without touching?

The answer to this would depend on the model of reality that you are using.

1) theoretical model - geometry - no. Two forms are either touching, or a distance separates them. If there is a distance, there are an infinite number of points in that distance.

2) theoretical model - physics for ordinary matter (solids, liquids, gasses): Touching here means that there is contact at the molecular level, and that that contact allows an exchange of energy (such as supporting weight or allowing electrical conductivity). Because of the electrical attraction of molecules, being closely adjacent would lead to touching (contact). Interestingly, though, close contact, as we see it, also has a lot of space. Otherwise, a metal block on a metal table would fuse to the table. In fact, if we create super-smooth surfaces at the molecular level, adjacent objects will make contact, then bond.

3) theoretical model - plasma physics. When a substance is in the plasma state, it is throwing off charged particles. As such, it has no clear boundary, and will make contact with anything adjacent to it.

4) quantum mechanics - at the level of sub-atomic particles, the picture changes. "Location" is no longer definable in any usual sense. Objects have no definable single location, only a cloud of probably locations (and associated probable velocities). And there is a gap in space between one quantum state and another. When a particle shifts quantum state (say an electron absorbing a photon) it "jumps" from one quantum level to another without going through the intervening space. This jump cannot be explained in any ordinary universe model where space is contiguous and touching (contact) is possible. So, in quantum mechanics, each particle is discrete (in a separate space) or united (in contact) with another particle. But we can't say exactly where it is!

5) electromagnetic waves and ocean waves. The notion of location does not apply exactly to waves of this form. But we do see the interactive effect of waves in the amplitude of the wave. These waves cross and yet do not affect one another.

6) sound waves do merge and blend on contact.