By Justin Melvin and Elizabeth French

These are NPL-style flats. They each have 14 words, and each has a base part that refers to one of the 14 senses of the adjective “flat”, as hinted by the title and the flavortext.

Their solutions are as follows:

Flat def. sense “Flat” base Other part of base Wordplay type Original flat
6 FIXED FAXED SECOND LETTER CHANGE Our hi-pri docs cannot be CLAIR. ’Til CHAIR, we must send mail by air.
2 HORIZONTAL ABHOR, CHORIZO, DENTAL TELESTICHAL ENIGMA Hate open crab? Use sausage anchor. With teeth, grin wide, Not up to rancor.
4 LEVEL OLIVE OIL CONSONANTCY Canola is not my suggested choice grease. Instead, extra-virgin GAS ups food’s taste GEESE.
9 LOW OWL HEAD-TO-TAIL SHIFT Keep barn CATSs SCAT on gloves, not sky. In jesses, birds won’t even fly.
12 MATTE MATTER CURTAILMENT The first two cans’re gloss, but then The pressing TEND: The third one’s TEN!
7 MONOTONOUS SUM REVERSED PALINDROME DELETION The drone goes on too long and DIVINE. In END, I execrate its whine.
8 OFF-KEY COFFEE SOUND SHIFT Beer, tea, black VIEW all serve to move A singer’s voice to sounding YOU’VE
1 PROSTRATE PROSTATE DELETION I do suggest for older guys To lay EYRES while doc checks your EYES
5 SHORT-HEELED SH or the E LED REBUS I’ve spools and kittens, leather pumps—I think that SOL shoes are for chumps.
3 SMOOTH MOOT TERMINAL DELETION They put up new rugs that are BLAH. That older ones were rough’s LA.
14 STABLE SEAT BELT LETTER BANK My horse should have a ROEHRICH? Fair. I’m clumsy on beasts from a CO-HEIR.
11 TAUT TAUGHT HOMONYM We’re BARE to string it BEAR to shoot, Avoiding your new shoes to boot!
13 TRACK RACK BEHEADMENT To PAT down vintners, one just scans For wine ATs detailed in floor plans.
10 UNINFLECTED UNINFECTED DELETION With negative culture and unchanging voice, I’m BACH and my BATCH dialect’s by choice.

Order by the sense of the adjective “flat” in (or the print 11th Collegiate Dictionary) and take the nth word of each flat, where n is the sense of “flat” in each flat. You get:

I open up my leather CHAIR
And move in my new third floor CO-HEIR

The cuewords clue this is a consonantcy. The solution is APARTMENT/PORTMANTEAU. Take the non-“flat” one to get the answer PORTMANTEAU.