As Robusto points out in comments beneath the question, there is no universally acknowledged rule governing whether to include or omit a comma after a conjunction at the beginning of a sentence. Robusto reports preferring to include such commas in academic documents, but many other writers and editors would not include them.
In my experience copyediting manuscripts for book publishers (including university presses) and later for magazine publishers, I don't recall ever having encountered a house style that required adding a comma after "And," "But," or the like. To the contrary, most house styles either said nothing at all on the subject or recommended omitting such commas, presumably for the reason that Words Into Type, third edition (1984) gives at the start of its long section on comma usage:
A comma should be used only if it makes the meaning clearer or enables the reader to grasp the relation of parts more quickly. Intruded commas are worse than omitted ones, but keep in mind at all times that the primary purpose of the comma is to prevent misreading.
The argument for including a comma after an opening conjunction is not, I think, grounded in a desire to make the meaning clearer (since the meaning tends to be quite clear without the comma, as Peter Shor indicates in a comment above), but rather in a desire to demarcate with exactitude the boundaries of the parenthetical expression that follows. Why Gregg Reference Manual would insist on such precision at the beginning of a sentence but not in the middle of one is a mystery to me.
There is nothing inherently wrong with using commas to break out parenthetical phrases regardless of where they appear in a sentence: It increases the number of commas in a work while (arguably) not making the sense of the text any clearer; but it's a style decision, and style decisions—if followed consistently—don't need to be justified.
On the other hand, if you don't want to add a comma after a conjunction at the start of a sentence, I don't think that you should consider yourself to be under any obligation to the preferences of Gregg Reference Manual unless your publisher has instructed you to obey it.
The Limited Power of the Comma
Before a coordinating conjunction joining two independent clauses, a comma only signals the beginning of the second clause. Its presence or absence, however, cannot signal whether a subordinate clause at the end of the sentence is to be distributed across both independent clauses or only one: that’s a question of semantics, not grammar.
Like a pronoun, a modifier following what it is to modify grabs on to the first logically available target and tends to stay there:
I chose a pottery class, but Robert signed up for creative writing because it was the only class without a scheduling conflict.
There is no ambiguity in this sentence despite the position of the subordinate clause modifier. Just as it attaches itself to creative writing and is perfectly happy remaining there, the subordinated clause restricts itself to the second clause 1) because there is no cue to read further to parse the sentence and 2) such a reading is semantically blocked by the verb chose, i.e., between at least two alternatives, while Robert’s crowded schedule only allowed him to register for one class.
Your example sentences, however, have been intentionally constructed to introduce ambiguity by subordinating a statement in the gnomic present (“Exercise is good”) or in the habitual present (“Whenever it rains”), both signalling a motivating truth or condition universal enough to be parsed across both independent clauses. Yet you place both in final position, where the tendency is to restrict their meanings to the second clause only. The presence or absence of a comma cannot resolve the ambiguity because it is a semantic one you’ve built into the sentence in the first place.
Topicalization
Consider these two pairs:
I swim every day because exercise is good.
Because exercise is good, I swim every day.
My friend plays tennis five times a week because exercise is good.
Because exercise is good, my friend plays tennis five times a week.
The inversion in italics gives the cause for the action slightly more topicality than the standard order in roman, which a writer might choose for a smooth transition to the next sentence or merely to vary sentence structure for better style.
If you wish to combine the first of each pair using a coordinating conjunction, however, the topicality becomes far greater because it is a cause common to both. Otherwise, why would you utter the sentence in the first place? The usual way to topicalize the cause is to place it at the beginning, just as one does for each person alone:
Because exercise is good, I swim every day, and my friend plays tennis five times a week.
or keep the standard order for the first clause only and let ellipsis take care of the rest:
I swim every day because exercise is good, and my friend plays tennis five times a week [because exercise is good].
or in some fashion reinforce the universal truth that exercise is good:
I swim every day, and my friend plays tennis five times a week because, as people always say, exercise is good.
Ambiguity resolved, but commas have little to do with it.
Your second example is similarly resolved by placing the condition first or by maintaining normal word order for the first clause:
Whenever it rains, I always feel fresh, but John feels somber.
I always feel fresh whenever it rains, but John feels somber.
Semantic ambiguity can only be resolved by a semantic solution, perhaps aided by comma placement, but don’t demand that a comma do all the work by itself.
Best Answer
The second version is a classical example of a comma splice, which some also consider a run-on-sentence.
It is not grammatically wrong (punctuation is not about grammar), but rather poor stylistically. Separate sentence should be separate, especially if your sentence already is long to begin with. The latter Wikipedia link also offers other remedies such as a semicolon or a coordinating conjunction. In your case, I recommend a simple period.