A practical tool that works well for me is to phrase the modifier as a question.
Case 1.
During a course of rigorous training, John Learned that the army was not for him.
In the above sentence, the opening modifier acts as an "adverb" modifying "Learned" THe modifier is not placed near to the word its modifying. Why is the above sentence grammatically correct? (i think it is)
Question: What happened
during a course of rigorous training? Answer: John learned something. This is an adverbial modifier, many of which can be thought of as modifying the whole main clause (i.e. it modifies "John learned" as a whole, not just the verb). In any case, adverbial modifiers don't have to touch the thing they modify as rigidly as noun modifiers do.
Case II
Sweating profusely, John Learned that the army was not for him.
Modifier seems to be correctly placed.
Question: Who was
sweating profusely? Answer: John. This is a noun modifier, as the answer is simply a noun. The modifier is placed correctly next to John.
Case III
As a chairman of the Fed, I often found that I was required to deal with international issues.
Modifies I or verb found ?
Question: (1.) Who was
a chairman of the Fed? or perhaps, (2.) What happened
as a chairman of the Fed? Answer: (1.) I was chairman, or perhaps, (2.) I found that.... This one is a little iffy, but I think it is an adverbial modifier like Case I, so I would go with question and answer (2.). However, even if you went with Q&A (1.), the placement of the modifier next to "I" would be fine.
The GMAT would provide more obvious errors such as "As chairman of the Fed, international issues often required my attention." The question-form method would help a lot, as neither "international issues" nor "international issues required" is a logical answer to any reasonable question drawn from this modifier!
Case IV
In the chapters that follow, I will set forth universal principles governing economics.
Does not modify I.
The modifier does not modify "I," but that's OK. The modifier is a prepositional phrase ("in"), which can be an adverbial modifier, unlike relative clauses ("which, that, who" etc.), which must be noun modifiers.
Question: What will happen
in the chapters that follow? Answer: I will set forth universal principles... The modifier modifies the action in the main clause, or simply the main clause.
SUMMARY:
Phrase the modifier as a question. Try to use the exact modifier phrasing within your question, and think about the logical, real-world question the modifier brings to mind.
If the answer is just a noun, then expect that noun to touch the modifier. These questions will take the form "Who was..."/"What was...."/etc.
If the answer has a verb in it, either alone or as part of a longer clause, then the modifier doesn't have to touch the verb. These questions will often take the form "What happened...?"