My Take the Sax is would be easier choice you can work around the embouchure.
I play both the clarinet and saxophone (Alto and Tenor) I struggled on the tenor for a few months due to the less firm embouchure needed on the Tenor Sax. but today the tenor is my first choice instrument...
Fingering is also much easier on the sax compared to the clarinet
Basic transpositions for sax,l trumpet etc
Bb instruments (Bb clarinet, tenor/soprano sax, trumpet ...)
Up one tone. C => D, F => G etc
Eb instruments (alto/baritone sax)
Down 3 semitones. C => A, F => D etc
Players of transposing instruments need to know these by heart (and being able to do simple transposition more or less by sight is also ...
If you mean that your written sax music will be in A, then you are not changing the key, as it would still sound in C. If you mean you want the song to sound in A, then that would put your sax part into 6 sharps!
Here's the quick dodge for transpositions (as long as you know your key signatures!) E♭ baritone has three 'built-in' flats. It's 'in E♭' after all! So, to play music for an instrument 'in C' (i.e. normal untransposed piano, no 'built-in' flats or sharps) we have to take away those three flats. Which is the same as adding three sharps.
It seems from your question that you want to play a song in C major on your baritone sax so you can play it in the same key as any accompanying instruments (that is, in "concert pitch"). To do that, you have correctly identified that you will be playing in the key of A major.
You can do this at sight by imagining that the treble staff's bottom line isn't ...
Saxophone are transposing instruments:
Your instrument is called Eb-Sax because it plays an Eb when you play a written C. This means it transposes a minor 3rd up (and as a Bariton an 8ve down!)
So you have to notate and play an A on your instrument when you want to hear C. (your instrument will transpose the A a minor 3rd up ...)
I would say Saxophone is the easiest instrument I have learned but the hardest to get a good sound on. The fingerings are easier than clarinet and the embouchure is easier than flute. The fingerings on saxophone are very similar to recorder (and flute) so they are very easy. Just try to get lessons if you can to avoid any mistakes while learning. Also alto ...