First of all, caringness isn't exactly a word, but if it was,
here are its rhymes:
actless actress aegis ageless aimless
airless Alice amos angus anise
anus anxious aptness armless artless
atlas auspice avis aweless axis
bacchus backless badness baldness bareness
baseless baseness basis basses beardless
bias bigness blackness blameless blandness
blankness bleakness blindness blitheness blondness
bloodless blouses blueness bluntness bodice
bogus boisterous boldness bondless boneless
bonus bootless boris boundless bowedness
brainless brakeless braless branchless brashness
braveness breadless breathless briefless briefness
brightness brimless briskness broadness bruskness
brusqueness buckleless bulbous burgess bus
business buss buttress cactus callous
callus calmness campus canvas canvass
captious captress carcass careless casteless
caucus causeless cautious ceaseless census
chalice challis changeless chasteness cheapness
checkless cheerless chicness childless chillness
chinless choiceness chorus Christmas churchless
cinderous circus cirrus citrus claimless
classless cleanness clearness closeness cloudless
coarseness codeless coitus coldness compass
concuss congress conscious conus coolness
coppice cordless cornice corpus cost-plus
costless countess countless coyness crassness
crestless crevice crewless crimeless crisis
crispness crocus crossness crudeness cruelness
cumbrous cureless curtness cuss cuteness
cutlass cypress Cyprus daftness dais
dallas dampness dankness darkness dateless
dauntless Davis deadness deafness dearness
deathless deepness deftness Dennis denseness
dermis derris dexterous dimness dingus
direness discus discuss ditchless doneness
doris doubtless Douglas drabness dreamless
dripless dryness duchess ductless dullness
dulness dumbness dustless earless Elvis
empress endless eros estrous estrus
eyedness eyeless faceless fadeless faintness
fairness faithless falseness famous fastness
fatness faultless faustus fearless fellness
fenceless ferris ferrous fetus fewness
fibrous fierceness fineness fireless firmness
fitness flatness flawless fleetness flightless
flushness focus fogless fondness footless
forceless formless fortress foulness foundress
fracas frankness freeness freshness friendless
fringeless fruitless fullness fungous fungus
furnace fuseless fuss gainless gameness
gaseous gaucheness gauntness gayness gearless
Genghis genus gibbous glabrous gladness
glibness glottis glumness goddess godless
goodness gorgeous gouges graceless gracious
grampus grandness gratis graveless graveness
grayness greatness greenness greyness grievous
grimace grimness grossness groundless gruffness
guileless guiltless gustless gutless hairless
haleness handless hapless harass hardness
harmless harness harshness hatless headless
heartless heatless heedless heelless heinous
heiress heirless helpless highness hingeless
hoarseness hocus homeless hoopless hopeless
horace hornless horseless hospice hostess
hotness houseless hubris hugeness humorous
humus huntress hybris hydrous ibis
ictus illness iris isis isthmus
janus jaundice jawless jealous Jesus
jewess jobless jointress joyless joyous
Judas juiceless justice justness kansas
keelless keenness keyless kindness kingless
kinless koumiss kumiss lameness landless
larcenous largeness lateness lattice laundress
lawless laxness leafless leanness leaseless
ledgeless legless leprous lettuce lewdness
Lewis lifeless lightness likeness limbless
limbus limpness lineless lipless listless
litheness litmus loathness lockless locus
loftless loneness looseness loris lotus
loudness Louis loveless lowness luckless
lupus luscious lushness lustrous lysis
madness madras magus maimedness maleness
malice manless mantis marquess massless
matchless mattress meanness meatless meekness
Memphis menace midas mildness mindless
minus mirthless mistress modeless modus
moistness monstrous moonless Morris mortis
mortise moses mucous muss muteness
mutinous nacrous nameless napless narcous
naris nauseous nearness neatness necklace
needless nerveless nervous newness nexus
niceness nimbus nisus nitrous noiseless
nonplus Norris notice novas novice
noxious nudeness numbness oddness office
ogress oldness oneness onus opus
oris ornice orris painless palace
paleness pappose pappous pappus Paris
parlous parvis pathless peeress peerless
pelvis penis percuss pertness phallus
phyllis pinnace pious pipeless pius
placeless plainness planless plexus plumpness
plus pocus pointless pompous porous
porpoise poshness poultice practice precious
preface premise priceless priestess primness
princess progress promise promptness proneness
proudness prowess ptosis pumice purchase
pureness purpose pus quaintness queerness
quenchless quickness rankness raptness rareness
rashness raucous rawness realness rebus
reckless rectus redness reinless responsibleness
restless rhesus rhombus richness rictus
rifeness righteous rightness rimless rimous
ripeness rodless roofless rootless roughness
roundness rubus ruckus rudeness rufous
ruleless rumpus rungless runless ruthless
sadness safeness sageness saltless sameness
saneness sapless sarcous scabrous scaleless
scantness scarceness scarless scentless scirrhous
scirrhus sclerous scoreless scrumptious sculptress
seamless seamstress seedless seeress selfless
senseless sepsis serous service sexless
shadeless shameless shapeless sharpness sheerness
shiftless shoeless shoreless shortness shrewdness
shrillness shyness sickness sightless sinless
sinus sireless situs skewness skinless
slackness sleekness sleepless sleeveless slickness
slightness slimness slowness slumbrous slyness
smallness smartness smokeless smoothness smugness
snideness snugness softness soilless solace
solstice solstices songstress soreness soulless
soundless soundness sourness spaceless spacious
spareness sparseness specious speechless spineless
spinous spotless spouseless spryness squareness
squatness stainless staleness stalkless stannous
starkness starless stasis stateless status
staunchness steepness stemless sternness stiffness
stillness stoutness straightness strangeness strapless
stratus strictness stringless stripeless strumous
styleless stylus stypsis suaveness sukkoth
sulcus sumptuous sunless sureness surface
surplice surplus sweetness swiftness syphilis
tactless tailless tallness tameless tameness
tarsus tartness tasteless Taurus tautness
taxless tearless temptress tennis tenseness
termless terrace terseness Texas thankless
thesis thetis thickness thinness thomas
thoughtless thriftless thrombus thus thymus
tideless tightness tigress Tigris timeless
timeous tintless tiredness tireless titus
toneless tongueless tonus toothless topless
tortious tortoise torus toughness traceless
trackless tractless traitress treasonous treeless
trellis trespass trimness trismus triteness
troublous trueness truss truthless tubeless
tuneless typhus us useless vagueness
vainness valveless vastness Vegas Venice
venous venus verbless vicious viewless
vileness villous vinous virus viscous
viscus voicedness voiceless voidness voteless
vulgus wageless waitress walrus wanness
warless waveless weakness wealthless weightless
weirdness wetness wheelless whiteness wholeness
wideness wifeless wildness windlass windless
wingless winless wireless witless witness
wondrous wordless workless worthless woundless
wrongness wryness xmas zealous