False Alarm? Not This Time! solution
by Kegler

CHICKEN LITTLE (22 Across) is generally considered an alarmist, but in this case “THE SKY” really is falling; six Down entries each lose one of those letters to the bottom row.

False Alarm?  Not This Time! solution grid

Across
1AL(UM)NA (ALAN anag)
5MEAT PIE ("MEET PI" hom)
12CAS(CAD)E
13RAN + T
14O + BEY
16MAN(T)IC
17R + ARE
19VINE (hidden)
20BETA (BEAT anag)
21ERICA (I CARE anag)
22CHICKEN LITTLE
25P(AGE)D
29ROOT (2 defs)
30ALAE (1st letters)
31SEllER
32T(E)ASER
33I'M + RE
34T(E)AR
35SNOW PEA (WEAPONS anag)
36ESTATES (TEA SET'S anag)
37conSTRICT
Down
1ACORN (NARCO anag)
2pUSHERS
3NA(I)VE
4AD + MIRE
5MEANINGLESS ("LEANING MESS" spoonerism)
6ERNE ("EARN" hom)
7AN + TRA (ART rev)
8TRI(BUT)ES
9P(ACK)E + R
10INS + TALLER
11E + THANE
15BA(THRO)BES (THOR anag)
18ET(C)ET + ERA (TETE rev)
22C(nUTs)RATE
23LEA + RNS
24GO + TAT
25P(LANE)T
26DESPOT (2 defs, pun)
27TEMP + I
28TR(EAT)Y (TEA anag)
30A + SHE