Since quite some time I get the impression, that especially in US-American, spoken english the pronunciation of 'e' (as in 'best' — /ɛ/) seems to shift towards 'a' (as in 'flat' — /æ/). Some recent examples:
Is it just my impression or is this a trend? Or is it simply me being more often exposed to mostly US-American accents that have been pronouncing it that way ever since?
In simple words, when [ɛ] is lowered, it becomes [æ]. It's what's happening in the examples the OP has given. There are a few factors that are responsible for this shift. Two of them are:
[ɛ] is an open-mid vowel while [æ] is a near-low vowel. One of the reason that accounts for the lowering of [ɛ] to [æ] is the 'Northern Cities Shift' from the Inland North dialect of the United States (as Legatrix pointed out in their answer). It's a chain shift of vowels through which some vowels have changed their positions in Inland North dialect region.
Another factor is the 'California Vowel Shift' through which some vowels have been shifted in Californian and other Western American accents. So you're hearing them correctly.
From Wikipedia:
California Vowel Shift: [...] /ɛ/ is pulled towards [æ] (wreck and kettle are sounding more like rack and cattle), /æ/ is pulled towards [ɑ], and /ɑ/ towards [ɒ] (cot and stock are sounding more like caught and stalk).
Here's the vowel chart of California vowel shift (CVS) where [ɛ] is indeed lowered to [æ]:
This paper has summarised CVS as follows [modified]:
There are also other mergers such as the pin-pen merger, which is a merger of /ɛ/ and /ɪ/ before nasals ([m, n, ŋ]). The resulting vowel is usually raised. Homophones include pin-pen, meant-mint, tent-tint etc. It's found in Southern and Western US.
I'm not very well-versed in American accents, but from what I've read, these shifts occur in many American accents. Read the following papers for more details:
Here is the relevant paragraph from Donka Minkova's A Historical Phonology of English (2013).
(Note that the vowel you are talking about is /ε/, also referred to as the DRESS vowel.)
As with /i/, the historical ‘stability’ of the short mid front vowel becomes suspect if one looks into the PDE realisations of /ε/. A backing to [ʌ] or lowering towards [a] occurs as part of the Northern Cities Shift; it is also reported for the San Francisco Bay Area and Canadian English. Younger RP speakers are also producing more open allophones of [ε] (Hawkins and Midgley 2005). A strong tendency for raising of [ε], especially in the younger generation, characterises NZE (Trudgill et al. 1998), illustrated by the confusability of, for example, check-in-counter ~ chicken counter. Raising is also common in the Southern US dialects and in north-eastern Newfoundland (Thomas 2001: 18–19). Another allophone involves the insertion of a diphthongal glide, especially in stressed monosyllables before a voiced coda, so that in popular London bed, leg can have [-ei -], and in the American South it can be a fully diphthongal vowel, so that dead is pronounced [dej əd] (Thomas: 2001: 18–19).
