Your agent gave you incorrect information.
First and foremost, to be entitled to receive Florida Homestead Tax Exemption benefits in 2018, the following two deadlines must be met:
Must be a Florida resident and own/occupy the property as your permanent residence on or before January 1st, 2018; and
The Florida Homestead application must be received by the local Property Appraiser's office by March 1st, 2018.
So in your situation, the property taxes for 2018 will be based on the current owner's homestead exemption.
The sale of the property in January will, however, trigger the end of the current owner's homestead exemption as of 12/31/17. So you, as new owner, will need to file for your own homestead exemption for 2019 and beyond after the closing.
Please note that your homestead exemption will be not be based on the current owner's assessed value, but rather based on a new assessed value that takes into consideration your purchase price.
Since the homestead exemption caps property value increases at 3% per year, if the current owners have lived in the home for a number of years, the "market value" used for calculating property taxes may be significantly less than the property's actual value.
The property taxes due at the end of 2018 will be based on the seller's assessed value, the 2019 taxes based on your new assessed value. Odds are that your property tax bill will be increasing in 2019.
The Hillsborough county property appraiser's website offers you a detailed explanation of the above, go to their page and click on "property info" for the link.