"ㄴ after a verb and 스래 meaning" (스래 -> 스레. "스래" doesn't exist)
"하지 못한 고백을, 호근 고집스레 삼킨 아야기" (호근 -> 혹은: or; or else)
=> A story of a never made confession, or stubbornly swallowing it back.
The sentence is structurally flawed ("못한 고백을 ... 이야기" is a bad form).
One way to correct it is:
고백을 (차마) 하지 못한, 혹은 고집스레 삼킨 이야기.
=> A story of stopping just short of, or stubbornly swallowing back a confession.
고집스레 is "stubbornly", as you said. (Both 고집스럽게 and 고집스레 are adverbs for 고집스럽다(adjective)).
A verb ending in ~(ㄴ/은) is an adjective form denoting a past action or its result, similar to English's -ing form.
It is good to understand the class of ~(ㄴ/은/는) together, which differs depending on whether it's a verb or an adjective.
~는: ongoing (present tense) action. 식사를 하는 사람들 (people dining), 언덕에 올라가는 아이들 (kids going up a hill).
~(ㄴ/은): done (past tense) action. 식사를 한 사람들 (people who has dined). 언덕에 올라간 아이들 (kids who have gone up the hill).
~는: ~는 is not possible because adjectives by definition can't denote an action.
~은: present tense state. 큰 집 (a big house), 푸른 하늘 (blue sky), 젊은 사람 (a young person).
There are also other related forms: ~(ㄹ/을/를) (future), ~던 (past experience), ~(았/었)던 (distant past), etc.
These adjectival forms (like English participles) of verbs make an important part of Korean sentence structures.