「海で泳いだ。そしたら海豚を見たよ。」is correct. If you make it one sentence it would be 「海で泳いだら海豚を見たよ。」

Thank you for your corrections! So you are saying that 「海で泳いだ。イルカが見える。」　is better? I thought that イルカを見た　in this context could be like 私はイルカを見た。(I saw a dolphin). Since 見える is intransitive, wouldn't the sentence イルカが見えた。 mean "A dolphin is visible" instead of "I saw a dolphin." How would I use 「そしたら」to indicate that I did not expect to see a dolphin while swimming in the sea?

I think "swan" is a touching mistake of "swam". I think that the good translation of the sentence "I swam in the sea and then I saw a dolphin!" is 「海で泳いで（い）たら、イルカにでくわしたよ！」（When I was swimming in the sea , I came across a dolphin!） or 「海で泳いで（い）て、イルカがいるのに気付いたよ！」（I swam in the sea and noticed there was a dolphin!）or 「海で泳いでいる時、イルカを見たよ！」（When I was swimming in the sea , I saw a dolphin!）. Your「海に泳いだそしたら海豚を見たよ。」 is incorrect. Firstly "に" should be "で". Secondly "。" should be inserted between "だ" and "そ". Thirdly "を見た" should be changed to "が見えた". "そしたら" needs an unexpected affair. "見た" is your willing act. The English word "see" should be translated to "見える" in Japanese in some sentences like this.