Hi! :)

Suggestions:

설핏 스며든 체온: 설핏 is an adverb meaning 'dimly' in terms of sunlight, or 'briefly' in terms of a thought popping up. In this case, I think briefly would fit best; briefly permeating body heat. 체온: body heat/temp.
매일 난 익숙한 낯섦: Everyday I'm familiar unfamiliarity (낯섦 isn't exactly a correct word, but it's just 낯설다 + (으)ㅁ to make it into a noun).
너라는 날을 시작하겠지: I start a day called you
You could also translate it as 'Every day in familiar unfamiliarity I start a day called you', which I think may be more correct.
Note: 너라는: 너라고 하는. It's an adjectival phrase to describe or quote whatever comes in front of it, in this case it's 너.

내 삶이 그동안: In the meantime my life has been
방황하듯 찾은 지금: wandering around like I was searching for the present/now
내 존재의 이유는 너란 그 믿음: The belief that reason for my existence is you.

네 사랑은 하늘보다: More than the sky/heaven, your love
기분 좋은 바람을 몰고 왔었고: Brought a better wind
Altogether: your love brought a wind that made me feel better (기분 좋은) than the sky/heaven
네 사랑은 바다보다: More than the sea, your love
너그럽게 내 맘을 이해했었어: Tolerantly/leniently understood my heart/soul
Altogether: your love understood my heart/soul more tolerantly/generously than the sea

오래 그려왔던 휴식처: 그리다 has two meanings, one of which is 'to draw/paint/depict' and the other, which is 'to yearn/long for, to miss'. The latter is meant here. "A resting place I've yearned for for a long time".

지쳤던 내 맘이 조율하듯: As though to tune/adjust (you could even translate it as 'recover' in this case) my heart/mind/soul which has become tired
맞춘 두 눈: You locked eyes with me (눈을/에 맞추다 would be more common, but this is phrased as 'two eyes that met/adjusted'), it's tricky to translate ngl.
넌 그렇게 매일
날 치유해 준 꿈: You're a dream that heals me like that every day.

Literally:
지금까지 날: Until now, me
사랑해줘서: For loving me
또 아껴줘서: For adoring me again
널 만나서 난: Because I met you I
행운이었어: Was my lucky
참 고마웠어: I was so thankful

You can translate it as: until now you love me, and adore me again. I was lucky to have met you. I was so thank/grateful.