SAVANNAH — Within two weeks, officials expect to sign a cost-sharing deal with the federal government that’s needed to start deepening the busy shipping channel to the Port of Savannah, the Georgia Ports Authority’s top executive said Monday. That means construction could begin within a few months on the $706 million project state officials have pursued for 15 years. “Everybody is elated that it now looks like the final authorization is going to happen,” Curtis Foltz, the port authority’s executive director, said after telling the agency’s board of directors he expects the agreement to be finished in “10 to 14 days.”