When a tranposing instrument pitched in E flat plays a written C, the actual note produced is E flat. So, a minor third higher or a major sixth lower or an octave or more away from that. Knowing that the baritone sax is pitched in E flat alone does not tell you which octave. You need to know. The alto sax is a major 6th lower and the baritone is an octave and a major 6th lower. So, you need to compensate for this shift by writing an octave and a major 6th higher.
Your music is in F. A major 6th higher is D so you should write the sax part in D. Your first note is F and you have correctly moved it up an octave and a 6th.
Actually, the two are linked. Since it opens on the tonic, figuring that the sax part opens with D tells you that the key is D.
Another way to get the key is to know that the sax will have 3 more sharps. Think of flats as negative sharps. Concert pitch is 1 flat so -1 sharp and the sax will have 2 sharps.