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The poskim in Orach Chaim 32 talk about the requirement that the letters for tefillin and mezuzos be written כסדרן, in order. As a result (32(23)), if a letter is missing there is no way to fix it later. But if a letter is extra, it can be erased, leaving the remaining letters in order.
However, if the erased letter is in the middle of a word, removing it divides the word in two. That can only be fixed if the previous letter can be stretched to fill the gap, like with a reish, see the nos'ei keilim there.
You can't stretch the following letter on the left, or other letters, because you'd need to erase at least one letter first to make way [or at least enough of it to ruin its tzurah: think of erasing the vertical of a reish in order to move it over and then stretch it back to meet the top], and then when it is rewritten we would be שלא כסדרן again.
They do suggest that sometimes you can "thicken" the letters before and after slightly, enough to fill in the gap so that the word no longer appears divided.
Of course, that won't help much if a couple of words were extra; you'll still have a big gap that looks like a פתוחה or סתומה that shouldn't be there. Again you're going to need some reishes and such (להדר"ת or whatever you hold may be stretched) in just the right places.
So I was wondering: Does anyone suggest the idea of shifting the letters bit by bit: thickening them a little on one side, then shaving them on the other, then repeating, being careful never to destroy their tzurah? It would seem to me that you could move letters or even words over that way, making way for the stretchable letters to fill the remaining gaps, without losing כסדרן.
Is this a) a known technique, b) works but it's way too much trouble to be worth it, or c) forbidden and if so why?

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This is a known technique. It's generally considered invalid and called פנים חדשות. See Lishkat HaSofer 9:6:

ונ"ל פשוט דאם לאחר שהעבה איזה אות מצד זה גרדו מצד השני ולא נשאר רק הכתב החדש הוי שלא כסדרן כיון דליתא לכתב הראשון‏
...and it seems obvious to me that if after he thickened some letter from this side he erased from the other side such that only the new writing is left, it is considered out of order since the original writing is not there

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  • Another common case where this question comes up besides extra letters is switching the right yud between backwards and forwards in a tzadi or alef. – Double AA Jan 14 at 14:16
  • Tzaddi I know about, but there are STA"M scripts where they write the top yud of an alef backwards? – Meir Jan 14 at 17:38
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    @Meir For one, there's Ktav Kratshin hebrewbooks.org/pdfpager.aspx?req=1302&st=&pgnum=118 but anyway my point stands even if you just wanted to "fix" an accidental one. – Double AA Jan 14 at 17:59
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    @Meir Incidentally, having a zayin shape on the right of the alef (head overlapping both directions) is actually not so rare in pre 20th century non-european scripts – Double AA Jan 14 at 19:15

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