In 1829 he was crowned the "king of Nordic poetry" (by Esaias Tegnér, who would be his Swedish parallel) in the cathedral of Lund, Sweden, based on a vast production of poetry, theatre plays and prose, inspired by Goethe, Fichte, and Schelling. See also Baggesen.
He wrote the song "Der er et yndigt land", which is now the national anthem of Denmark.
The Internet has very little information in English about Adam Oehlenschläger. Some exceptions are Slider Encyclopeadia and Columbia Encyclopedia, 6th edition. Several presentations in Danish can be found from Project Runeberg and e-lysium.