Cry it out method

Okay so first i want to get made clear that i do not mean ignore my babies cries all night!! I’ll explain my situation and please i would love your input. My daughter is 3 months and extremely needy. I would consider her colic because she is crying and upset constantly. She is rarely happy. I’ve got to the point where now im running off 1-3 hours of sleep a night and its not one stretch its 20 min here, 40 min there. Im so exhausted to the point i cant even keep my eyes open. Im so overwhelmed with my baby and her needs. I cry all the time and it causes me more stress then i can physically and mentally handle. My baby will sleep at night but wakes 4-10 times. When shes up, its for an hour or 2. Sometimes even 3. I will feed her and rock her back to sleep and as soon as i put her down all hell breaks loose. She just wants to be held. I CANT lose the very little sleep i get to holding her all night. She still eats 2-4 times in the night and i wont deny her that obviously but once shes done eating and I’ve put her to sleep is it okay to let her cry it out? Ive done it a few times out of frustration and she went to sleep within 10 min or so. I just feel guilty. I don’t know what to do and i feel so defeated. I have no help so it’s just me