The Mishne mentions those who bring an oleh v'yored. Amongst them are shmiyas kol and bitui sefasayim. This means someone one who swears that he doesn't know eidus but he really does and someone who swears that he doesn't have someone's object but he does. Now he admits that he was wrong. I always thought that they bring an asham. Looking at the posuk in Vayikra 5:6, however, I see that the words v'heivee es ashamo.... and l'chatas are mentioned. Also it's a female sheep or goat which is usually a chatas (kisbah). Which one is brought - a chatas or an asham?
The Shevu'ah you mention, when someone denies having a stolen object, is Chayav Asham as you write. However, that is not a Shevu'as Bituy, but rather a Shevu'as ha'Pikadon. A Shevu'as Bituy is "l'Hara O l'Heitiv," such as a Shevu'ah "she'Achalti" and "she'Lo Achalti," or "she'Ochal" or "she'Lo Ochal." That Shevu'ah is Chayav an Oleh v'Yored (which is a Chatas). "Ashamo" is the stolen article.