How to prove NP-hardness from scratch? I am working on a problem of whose complexity is unknown.
By the nature of the problem, I cannot use long edges as I please, so 3SAT and variants are almost impossible to use.

Finally, I have decided to go for the most primitive method -- Turing Machines.

Oddly enough, I could not find any example of NP-hardness reduction done directly by modeling the problem as a language, and showing that a deterministic Turing Machine cannot decide whether a given instance belongs to that language (I might've messed up with the terminology here).

So, assuming that there are no problems to perform an NP-hardness reduction, how does one prove that a problem is NP-hard? Are there any publications that does this?

I also want to add that I know how to perform an NP-hardness reduction. However, the problem that I am tackling is a "localized" geometric problem, that it does not allow me to model any given instance of 3SAT, 3-coloring, vertex cover, etc.   

The immediate question comes to mind: "what if the problem is polynomial time solvable?"
Well, that is also a possibility, but I want to exhaust everything before I move onto designing an algorithm.
The only two methods I've seen are (a) a reduction or (b) direct proof (as in the proof of the Cook-Levin theorem).  It is almost universally the case that a reduction is easier than a direct proof.  Therefore, I suggest you keep trying to find a reduction, and consider other reduction partners.  There are lots and lots of problems known to be NP-complete; perhaps you can find one that is a more suitable partner than 3SAT.

Possibly useful: How do I construct reductions between problems to prove a problem is NP-complete?.