Just signed up to share some of the solutions (3 of them) I was able to come up with in 6 steps:
OH MAN! After a lot of searching I was able to do it in 5 steps (5 solutions):
I confirmed the words on Anagrammer.
EDIT: although all words are accepted on Anagrammer, someone pointed out in the comments that two of the words used in the 6-step solutions ...
Here's one way. It probably isn't close to optimal. 42 steps if I've counted right.
Credit where due: the path from EIGHT to SEVEN is derived from Hunter's answer to an earlier puzzle, though not much of that answer remains in what I have there now. JonMark Perry spotted what in hindsight should have been an obvious improvement in the path from FOUR to ...
This is a proof that Omega Krypton's answer of 38 is optimal, as long as these words are nonexistent (single implication):
First, a calculation of the theoretical best (although no suitable words may exist), by using a Levenshteins distance (word difference by letters) ...
In this answer I will show that five steps is the minimum number required (from the answer by @JoãoBravo).
Suppose by contradiction that there are four. Then the sequence will be of the form
F i v e, _ _ _ _, _ _ _ _, _ _ _ _, F o u r.
If the first letter remains unchanged the whole way through, the sequence is
F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r....
As @RossMillikan pointed out correctly, the proof from @TheSimpliFire is incomplete.
The following part
F i v e, F _ _ _, F _ _ _, F _ _ _, F o u r.
Therefore, at each step, one of i v e must be changed to match o u r, without any deviations. This is clearly impossible.
can be done with any of the following sequences
Five => F_ve => F_ue / F_vr => F_ur ...