Okay, I understand if the following word begins with a consonant, you use “a” and if the following word begins with a vowel, you use “an”. But what if the following word is some sort of abbreviation? For example, I might shorten “Lunar Module”, if I’ll be using it a lot, as “LM”. Now, the letter “L” is a consonant, but since it’s pronounced “el”, I’ll plant an “an” in front of it because it begins with a vowel sound.
Have I been doing this correctly? 
Yep.
To be clear, the rule is if the word starts with a vowel sound (and semi-vowels or glides like “y” don’t count.) Hence, it’s “a unicorn” not “an unicorn,” even though “unicorn” begins with a vowel.
Thus, your interpretation of the rule in accordance with initialisms or abbreviations is correct.
To add confusion, in the example you gave LM would probably be spoken as “lem.”
As the others have said, OP, you’re doing it right, although if you discarded “an” altogether and just used “a” for everything, you would be joining a pretty large group of people. It’s amazing how many people appear to have forsaken “an” altogether, even in formal speech settings such as newsreading.
I actually started a thread complaining about the issue almost a decade ago.
Partly because the official term was originally LEM, for Lunar Excursion Module. In later years, it was regularly shortened to LM, but the pronunciation stuck. If I were more used to LM though, I’d probably go with “el-em.”
The Command Module was always referred to as a CM, so no such choice there! 
Another rule of thumb: You should use “an” if a word starts with a silent consonant. For example, “He’s an honest man.”
This becomes tricky when different varieties of English are involved. For example, a Brit would probably say “Basil is a herb,” where I as an American would say “No, it’s an herb.” He pronounces it with an “H,” I don’t.
OK, that makes sense. But if you didn’t know that, then “el em” would certainly be the default pronunciation.
Some work of literature contains the sentence “The law is a idiot, a ass.”
You don’t really need a separate rule for that. The vowel sound rule covers it.
Mr. Bumble says that in Oliver Twist. But it has to be read in context. Mr. Bumble himself is a idiot, a ass. He’s not an educated man and doesn’t speak like one, except for his pretenses to loftiness.
Is it true that Brits pronounce the H in herb? I thought they tended to not pronounce the H when it’s the first letter, while Americans usually do. How did this one get flip-flopped?
It’s true. We definitely say Herb garden, rather than the ridiculous (to our ears) 'Erb garden.
Why different? No idea.
To quote Eddie Izzard, “Because it has a fuckin’ ‘H’ in it”.
According to this, it’s not so uncommon for words borrowed from French to be more anglicized in British English than American English. See also “fillet”.
The usage that drives me batty is the recent trend for otherwise educated people in the U.S. (including politicians and newscasters) to say “an historic…”
If you’re pronouncing the “h” sound (which these people are), it should properly be “a historic…”
All I can figure is that someone got it in their head that that the first incorrect usage was preferred, and others have followed, lemming-like (rather like the word “normalcy” instead of “normality”).
Yes–sometimes you have to be familiar with the speech of a particular discourse community to know which way to go. If your interaction with that community is only through text, you might not know when an initialism is spoken as an acronym. Everyone knows that in common speech UCLA is an initialism. So you know to say, “She’s a UCLA student,” rather than, “She an UCLA student.” But what about the (less known) Utah Vocational Academy (or UVA)? If everyone there calls the school /uvə/, then they say (and write), “She’s an UVA student,” but you might not know that, and write “She’s a UVA student.”
FWIW, President Harding didn’t coin the word “normalcy”; he wrote “normality” in his speech but read it as “normalty” and “amused reporters” changed it to "normalcy. " (Francis Russell, The Shadow of Blooming Grove: Warren G. Harding and His Times, etc.)
And I would say ‘an hotel’ whereas you would probably say an hotel.
I would always put ‘a’ in front of ‘U’ = yew. A university student. A Ucla student. An Uncle. A Uniform.
Quite true. However, if you’re using the longstanding acceptable practice of NOT aspirating the “H” in an unaccented first syllable, then the correct pronunciation is “an istoric.” Unfortunately, that is properly written “an historic” but folks who haven’t been taught the traditional pronunciation (drop the h) commit the mistake you (and I) lament.