To keep your ㅅ from leaking, the instructor says you need to focus on pronouncing the vowels to which the ㅅ is attached, rather than on pronouncing the ㅅ. You can do this by pronouncing the vowel sounds more slowly and more precisely. She suggests you could try first pronouncing just the vowels, without the ㅅ, to help you keep your focus on them. When you pronounce the word 소송, for example, first say slowly and clearly 오옹, and then add the ㅅ to pronounce 소송, while remembering to keep the focus on the vowel sounds. If you do this, the problem should solve itself.
The instructor also says some Koreans have a similar problem pronouncing syllables with the consonants ㅈ and ㅊ, a problem that can also be solved by simply focusing more attention on pronouncing the vowels to which they are attached. For example, when pronouncing the word 시작, she suggests stretching out the vowels sounds to help you add focus to them, such as saying 시작 as /시이자악/.
So, that is how you stop your 시옷 (ㅅ) from leaking.