Season one ended with the main characters leaving for some unexplained adventure. But in season two, they're still hanging around the capital. Does season two take place before the end of season one?
Actually, there is no season 2. Those 4 episodes were more like an OVA that Netflix mislabelled as season 2. Season 2 is coming, nobody actually knows the date when it's going to be aired.
What Netflix labeled as "Season 2" isn't exactly a new season, but rather special episodes from the Manga that aired on TV in Japan.
It seems like a continuity issue, to me. You are correct that at the end of Seaon 1, Elizabeth jumps on the Giant Boar and they all ride off, presumably to find Escanor the Lion's Sin of Pride.