Closest Woodall Prime

Brachylog, 28 26 bytes

:K≜+ℕ~lṗ.+₁~×[İ,J]h:2^₍J≜∧

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This is too slow for TIO for inputs bigger than 4.

Explanation

:K                             The list [Input, K]
  ≜                            Assign a value to K (0, then 1, then -1, then 2, etc.)
   +ℕ                          Input + K >= 0
     ~l .                      Output is a number of length (Input + K)
       ṗ.                      Output must be a prime number
         +₁                    Output + 1...
           ~×[İ,J]             ... = İ × J, İ being an integer...
                  h:2^₍J       ... and J = 2^İ
                         ≜∧    Check that it is possible to find a value for İ that satisfies
                                 all of those constraints. If not, go back to the beginning
                                 and try another value for K through backtracking.

We use a common trick in Brachylog for "find the closest X to Y": labeling a completely free variable (which we do when we use ≜ on the list [Input, K] will label it with this pattern: 0, 1, -1, 2, -2, 3, .... Therefore, when backtracking if the value Input + K doesn't work as the number of digits, the next choice we will try will be the next possible closest choice.