Short answer:
The vowel is usually /ɑː/ in both American and British English. The /l/ is pronounced roughly half the time in American English, but usually not pronounced in British English.
For more discussion, see below.
Two different transcriptions for the same pronunciation
Different linguists transcribe the same pronunciation in different ways. This is not always because they disagree about how the words are pronounced, but because they follow different principles in their transcriptions.
For example, the DRESS vowel is sometimes transcribed /ɛ/ and sometimes /e/. The symbol ɛ is closer to the actual phonetic pronunciation, while the letter e is more basic and easier to work with on computers. How do we choose which one to use?
Some linguists follow the principle of using the simpler symbol when there's no need to draw a contrast between the two, so they go with /e/. Others go with /ɛ/ because they prefer to use a symbol which is closer to the actual pronunciation. Using /ɛ/ is also common in comparative transcriptions – when transcriptions of English are side-by-side with those of other languages.
As a result, some dictionaries transcribe bed as /bɛd/ and others /bed/, but that doesn't mean these dictionaries disagree about the pronunciation of the word.
Vowel length is not always transcribed in AmE
You're running into something similar. Some linguists, in transcribing American English, prefer not to indicate vowel length. This follows Kenyon and Knott, who write in their book A Pronouncing Dictionary of American English (§55):
As a rule the length of English vowels is not indicated in the vocabulary. In few cases in American English as a whole is time length, or duration, of vowels significant—that is, used to distinguish from each other words otherwise alike. Instances are found here and there, some speakers distinguishing halve hæv from have hæːv by a longer æ, or vary vɛːrɪ from very vɛrɪ by a longer ɛ.
And so in their dictionary they generally do not include vowel length markers in their transcriptions, because it is unnecessary and this makes the transcriptions simpler. Some other dictionaries follow this convention in transcriptions of American English as well, and some do not, but in transcriptions of British English it is standard to include vowel length markers.
That is why you can find some dictionaries which write /ɑ/ in AmE and /ɑː/ in BrE, and other dictionaries which write /ɑː/ for both. Your ear is correct; the sound is [ɑː], even if it's sometimes written as /ɑ/ without a vowel length marker.
How should you pronounce the vowel?
The most common pronunciation is [ɑː] in both American and British English, and this is the pronunciation that I would recommend a learner of English target in their own production. Not all native speakers pronounce it this way, however. You may hear [æ], [ɒ], or [ɔː] instead.
Should you pronounce the /l/?
In the Longman Pronunciation Dictionary, John Wells includes the results of a Pronunciation Preference Poll for both American and British English:
American English speakers are split, with 53% preferring to pronounce the /l/ in palm, but only 15% of British English speakers prefer to pronounce the /l/.
Although there is no similar chart for calm in the dictionary, you can find a similar set of variations in its pronunciation, with some speakers pronouncing the /l/ and others not.
For learners targeting British English pronunciation, you should probably avoid pronouncing the /l/. Learners targeting American pronunciation can pronounce it either way.
