"Again, you have heard that it was said to the people long ago, 'Do not break your oath, but keep the oaths you have made to the Lord.' But I tell you, Do not swear at all: either by heaven, for it is God's throne; or by the earth, for it is His footstool; or by Jerusalem, for it is the city of the Great King. And do not swear by your head, for you cannot make even one hair white or black. Simply let your 'Yes' be 'Yes,' and your 'No,' 'No'; anything beyond this comes from the evil one." Matthew 5:33-37
In the Old Testament, how binding an oath was was judged by how closely it was related to God's name (Yahweh). Swearing by heaven or earth was not really binding, nor was swearing by Jerusalem. However, swearing toward Jerusalem was binding, because it was swearing towards the city of God.
If you have to swear for someone to believe you, then you probably haven't proven yourself to be trustworthy in the first place. Instead of swearing to prove your honesty, just be honest. Your reputation will speak for itself, and those who truly know you will believe you, even without an oath.
If you have been dishonest toward someone and have not yet sought forgiveness from them, find them, confess to them, and ask them for their forgiveness. They may not like you, and they may indeed hate you, but at least they'll know that you tried to be honest with them, even if it took you a little while.