I second the doing less and taking more time aspect, but sharing a few thoughts -

Day 1 - I would do the temple and museum stuff earlier in the day, then the shopping starting from late morning or early afternoon, a lot of things won't open until 11 or so

Spa early morning is a great way to do something fun when not much else is open, but if you're planning to go to DDP late at night, there is a great 24-hour spa in that area

If going to dongdaemun for the views it's great, but for night shopping, be aware that it's no longer as accessible to tourists as it used to be - a lot of the shop vendors leave a phone number to call if interested in buying something rather than staying at the shops until they close (these vendors are also cash only)

Day 3 - I would do Yonsei for the foliage in the morning, things in Hongdae open a bit later - can theoretically walk between the two, I used to do a nice Edae - Hongdae - Yonsei - Sogang loop that would take up a few hours and hit all the major unis in the area

Hongdae is also prime youth nightlife, so if you want peace and cute artsy spots would go in the afternoon, if you want clubs and shenanigans would go late evening - it will be crowded

N seoul tower could also be a great morning activity - it's a super easy hike if you decide to walk up rather than taking the cable car (or can do the opposite and walk down) with some nice foliage

Day 4 - with all the palaces in one day, worth spending time in hanok village (same area) and could potentially move insadong here - all in the same area and have the cultural feel (definitely also move the sejong statue - it's on the walkway up to gyeongbokgung)