https://www.mangago.me/read-manga/the_abandoned_empress/
I think this is the one you're looking for with the girl running to hug her dad but be warned Male lead is the shitty blue haired emperor and the story gives the excuse of "they're in love" without ever answering why. (lot of readers would've prefered kinda yandere green haired guy or boring red haired guy.
They explain the reason the male lead was an asshole was because of his mommy issues (his mom is dead but he considered MC's mom kind of his mom because she watched over him when he was younger so when MC was born he was jealous that she got her own mothers attention more than him) and daddy issues (his dad was kind of distant but was very proud and praised MC more than ML)
Hey hey I’m trying to find a webtoon where the FL is a seamstress and ML falls in love w her at first sight
She fixes his button
Also one where she wakes up from past life and runs to hug her dad at the training grounds?
Thank you!! (๑•ㅂ•)و✧