Despite what the Illinois attorney advised, a lease in Texas can be modified orally. The lease can be modified orally even if the lease states that there can be no oral modifications. The lease can be modified orally even if the lease states that any changes have to be in writing. Texas courts have held that the parties can orally agree to waive provisions requiring written changes or that there be no oral changes.
The problem with oral agreements or changes is proving them. In your instance, there does not appear to be an oral agreement that specifically dispensed with the 30 day notice provision. You appear to back into such an agreement through an omission.
Now, if the whole process appears to be deceptive or unconscionable, the storage company may have committed a violation of the Texas Deceptive Trade Practices Act, Texas Business & Commerce Code sections 17.41, et seq. ("DTPA"). You may be able to negotiate a satisfactory resolution by writing a letter by certified mail to the storage company providing notice of a violation of the DTPA through deceptive or unscionable conduct. If you were to prevail in a DTPA claim and the violation was knowing or intentional, you could recover up to three times your damages.
You will probably need an attorney to help you with the DTPA notice letter, but that will probably eat up the amount in dispute.