No worries! Yes, the monsoon season rains can start in the last two weeks of June, irrespective of what locals think the official start day should be
If you're going in October, this changes everything. You can likely get away with still wearing shorts to mid-October, but the sweltering heat is largely over. This will make touring around easier, from walking to the subway. Day 3 of your itin has you doing the Namsan-Itaewon-Gangnam run. Be mindful this is a lot of time on the subway from Itaewon to get down to Gangnam and that line can get jam packed busy. Likewise for Busan, as the run from Haeundae to Nampo-dong shopping district can take awhile, let alone the airport or Busan Station for the KTX.
Most importantly for surfing: you stand a far better chance of a Japan and/or China tracking typhoon in Sept-Oct generating swell that could make surfing possible in Busan, but especially on Jeju. Be warned: flat conditions can give way to powerful swell pretty quickly depending on the track of the typhoon. Don't underestimate it. Unfortunately, a Korean bound typhoon could complicate your travel plans. Also be mindful there are two public holidays in October, which will make tourist sites more busy and KTX, plane tickets more expensive.
Enjoy eh!!