Suggestions for a SC trip

Hello everyone,

I'm planning a 10 nights trip to South Korea between 25 July - 05 August.
Since I know it is extremely hot, I was planning to pass 6 nights in Seoul and 4 nights in Jeju Island to enjoy the beaches and the Island itself.

To save some time in airport exchanges, I was thinking about this route:
26 Jul - Arriving at Incheon airport, pass the night in Incheon.
27 Jul - Departing for Jeju from Gimpo airport
31 Jul - Returning to Gimpo and reach Seoul
5 Aug - Leaving Korea from Incheon

Do you think it is feasible or you suggest better routes?

The alternative trip would be Seoul - Busan via train and some one-day trip inbetween, skipping Jeju Island.

Do you think it is better to see Busan and other cities than Jeju for a first time in South Korea?