The shared theme is:
Pop singers or pop bands and their hits.
The twist is:
Each clue describes a singer or band as well as a song title. The first half describes the artist, the second half the song. That's why there are two eumerations for each song.
(The question says there is a twist on the "usual format", but several of these cryptics are quite loose. For example, the definition often contains wordplay and isn't more than an allusion.)
Rumor suggests a bug order, apparently including an annular celebrity, is
waiting for dawn. (7 and 4/5/3/3)
The Beatles, Here Comes The Sun
BEATLES, homophone ("rumor suggests") of BEETLES (bug order).
RINGO (RING + O, both "annular") STARR ("celebrity") was one of them.
HERE COMES THE SUN ("waiting for dawn")
Half a GPS, snide - associated with some cads - is as in orbit. (3/5 and 4/6)
Tom Petty, Free Fallin
TOM(tom) ("half a GPS") + PETTY ("snide")
Tom Petty's band are the Heartbreakers ("cads")
FREE FALLIN' ("is as in orbit")
French dear who previously partnered with a non-elderly guy, not so incredibly, telephones pollinator's departure. (4 and 7)
CHER ("dear" in French), previous partner of Sonny ("non-elderly guy").
Homophone ("telephones") of BEE LEAVE ("pollinator's departure") - also hinted at in "not so incredibly".
German, novice bird of prey, wanted composer's interlude to help him drop off? (5 and 4/2/7)
Falco, Rock Me Amadeus
FALCO(n) ("novice" = "beginner" of "bird of prey" or FALCON)
Falco was Austrian and many of his songs are in German.
ROCK ME ("help him drop off") + AMADEUS ("interlude" or middle of the composer Wolfgang Amadeus Mozart)
"Experienced," I heard first of RLS, along with a chick's mother and outputs
from a leaky faucet, gave forth an unusual cloud. (4/7 and 6/4)
Jimi Hendrix, Purple Haze (found by jafe)
Homophone ("I heard") "first of" JIMMY [legs] (restless leg syndrome, "RLS") + HEN ("chick's mother") + DRIX (sounds like drips, "outputs from a leaky faucet"?)
Jimi Hendrix's band was The Jimi Hendrix Experience, hence "experienced".
PURPLE HAZE ("unusual cloud")
A group of women, moving twice, is good at keeping secrets. (2-3 and 3/4/3/6)
The Go-Go's, Our Lips Are Sealed
GO + GO ("moving twice") + S. The rest is rather straightforward.
Blended pitiful detergent lump, promoters of skateboarding and motocross, turn traitors for auto commercials. (1/11 and 9)
X Ambassadors, Renegades
Anagram "blended" of SAD BORAX MASS ("pitiful detergent lump"), also X (extreme sports such as "skateboarding and motocross") AMBASSADORS ("promoters").
RENEGADES ("traitors"); the song was written for and heavily used in ads for the Jeep Renegade.
City close to someone from south of the border, employing brass, can't decide
between two possible times. (7 and 2/2/1/2/1?)
Chicago, 25 or 6 to 4
CHICAGO, just one letter away from CHICANO (Mexican person is "from south of the border" if you happen to be in the US). The band prominently used a brass section.
The title means 25 or 26 minutes before 4 a.m., i.e. either 3:34 or 3:35 (Using digits in the solution is quite unusual.)
Dreamy basic comment provides anti-war flavored soda. (3 and 6/5)
R.E.M., Orange Crush
REM (the BASIC keyword for a comment); REM is a dream phase.
ORANGE CRUSH (flavored soda). The song has an anti-war theme.
Reverse currency and reverse current, in reality sharing a name with a step by step mother pretender, ask a hypothetical question about being like Jesus. (5/5 and 4/2/5)
Eddie Money, Walk On Water
"currency" becomes MONEY and "reverse current" becomes EDDIE; then "reverse" the two to get EDDIE MONEY. His real name is Edward Joseph Mahoney, sharing a name with Suzanne Somers née Mahoney, who played the mother (and also stepmother, as it happens) on "Step by Step".
WALK ON WATER as Jesus purportedly did.