When administering an oath for an affidavit or another statement that requires the signer to swear or affirm the truthfulness of his or her statements, you may normally ask the affiant or signer to raise his or her right hand. You might then say:
"Do you swear that what you are signing is truthful?"
People often find this a little unnatural because to swear is to make a declaration to a deity.
If the signer is not a believer in deities, he or she will often refuse to swear. Some signers absolutely do believe in God but believe it is disrespectful to "swear to God" which is exactly what swearing is.
Make this a little easier to handle for such clients.
First of all, it's not required in Texas to ask the signer to raise his or her right hand, so consider cutting that out of your routine.
Other states' notaries must check their laws about this--I only speak to Texas's requirements on hand-raising to take an oath.
If a signer refuses to swear, it should not be a problem. Ask if he or she will "affirm" said document's content. If the signer asks what that means, you may explain that to affirm means to strongly and publicly assert something as fact and it doesn't include involving a deity.
Finally, note that you administered an affirmation in your journal.