It's just called alternate bass.
Regarding stopping the strings... well, practice! – that's actually a crucial ability to have on bass, in any genre.
I guess the problem you're referring here is specifically that you're always changing the string after every note, i.e. you can't just use the next-plucking finger to damp the previous note, as you would in (assuming you pluck with two fingers)
V:B clef=bass-8 transpose=-12
[V:B] "i"A,, "m"z "m"A,, "i"z "i"A,, "m"z "m"A,, "i"z
When switching strings, you'd instead damp with the previous-plucking finger:
V:B clef=bass-8 transpose=-24
[V:B] "m"D, "m"z "i"A,, "i"z "m"D, "m"z "i"A,, "i"z
This is done with a sort of rocking back and forth motion of the plucking hand.
Note that single-finger stopping generally doesn't work 100% – you don't so much stop the note as squelch it to an overtone. On option is to damp with the whole palm, but this is quite unpractical at tempo.
I personally don't generally bother with any of that, and just damp primarily with the fretboard hand. I hardly use open strings at all on electric bass, especially not in sequence. So for D-A, just play both of them on the A string, then you can easily damp with both hands. For E-B, I'd play them on the A- and E-strings on 7th fret, fingering with pinky and ring finger, respectively, while the index finger rests over the 5th fret.
An alternative, which is arguably more appropriate for folk, is to go with full double bass technique, i.e. you pluck most notes with the side (rather than tip) of your index finger. This is much more effective at damping undesired strings, especially the lower ones.