This is where enharmonic equivalents can get annoying! I agree with this:
...so (G# B D F) is more common,
as it is of course a common and very useful chord for modulating, particularly in the manner you described. This however:
and (G# B D Fb) is less common.
is actually quite common when you consider that it's actually an E Dominant 7th 1st inversion chord, but with the E "mis-spelled" as F-flat!!
If you are going to use a diminished 7th make sure at least one of the notes (most commonly the root) is a half step away from the next chord. <snip> make sure they will lead to the half step above the root very strongly.
Sage advice. Typically, a Diminished 7th can be used as a substitute for a Dominant 7th when modulating, eg, in C major, you could go to the relative minor (a min) by preceding it with a dim 7th on G#. You could also modulate to the Dominant (G) by preceding it with a dim 7th on F#, always assuming that the melody line allows it. However, the root of the Dim7th chord does not
always have to rise a semitone to the next chord, it can be equally effective falling as well.
In the major key where the melody permits, I sometimes like to use a dim7th falling to a 2nd inversion tonic, then to a Perfect cadence, this has the effect of tricking the ear a bit, and not going quite where it's expected. Again, in C major, an example would be:
G#-B-D-F --> G(nat)-C-E --> G-B-D (F) --> C-E-G
All this talk of diminished7ths however is somewhat moot in relation to your query. Luckington is a grand tune, and of moderate length, so one suggestion is perhaps not to try changing every chord! The version I've sourced is in D so note names used refer to that key. The passage you've asked about can't effectively use a Diminshed 7th because of the way the melody skips up the tonic chord. On "the" and "world" it's A and D respectively which plainly suggests a Perfect cadence. Even if you wanted to go to the relative minor (b min) on "world", you couldn't precede it with a diminshed 7th chord because of the A-natural in the melody.
The first attached graphic is the first thing I tried on that phrase, it's by no means definitive, but capable of spicing things up a little (you'll notice that traditional voice-leading always goes out the window with this sort of stuff!). It's an example of what I mentioned before about a diminished chord falling rather than rising, but in this case, to actually get a dim chord there mean't flattening the 7th of the key.
And just for a bit of fun, the second graphic has a mildly out-there suggestion for the final phrase (showing my fondness for the flattened 6th!)
Honestly, the best thing you can do is take some theory classes, or get a good textbook on harmony
Again, very wise words. The nearest in my bookshelf at work tend to be Walter Piston and Annie Warburton's respective Harmony books, but there are plenty of other good books out there. Read through exercises, and thoroughly analyse them to see how and why these things work.
Good luck!