For example, say Tennessee outlaws gasoline powered cars. A Georgia
official intends to drive a state-owned gasoline powered vehicle to
Kentucky on official business. Can Tennessee force them to go the long
way around, or does Georgia have sovereign immunity from Tennessee
law?
If Tennessee's law is valid, sovereign immunity for a Georgia official will not prevent Tennessee from enforcing the law against a Georgia official on official business within its territory.
As a practical matter, the Tennessee law may violate the "dormant commerce clause" or be pre-empted by federal laws relating to automobiles which related to interstate commerce, but if the Tennessee law is valid, sovereign immunity won't help the Georgia official who is physically located in Tennessee.
As a practical matter, there is almost no case law on point (some of the most pertinent cases are dicta in recent U.S. Supreme Court original jurisdiction cases between states such as one related to the 2020 election and one by Nebraska against Colorado related to marijuana being carried from Colorado where it is decriminalized to Nebraska where it is not), but this is the way that the law is lived and is very likely to be the way that a court would act.
Also, the immunity available to a Georgia official is not the same as the immunity available to the State of Georgia, which has constitutional protections under the 11th Amendment to the U.S. Constitution, that do not apply to the Georgia official acting in the Georgia official's official capacity. This distinction has evolved in civil rights and public law contexts not involving two states, but
the distinction should apply in the context of this question. Tennessee could cite the official for violating its law within its territory. But Tennessee could not fine the State of Georgia in that situation.