There is nothing wrong with using a capo, if it makes the sound you want.
Having said that, the barred shapes you are using are pretty simple ones, and it will be worthwhile learning to play them without a capo.
In your example you are raising the pitch of the whole song by 5 semitones.
- The Am becomes a Dm
- The E becomes an A
- The A7 becomes a D7
- The Dm becomes a Gm
Most people choosing to barre the shapes for this set of chords would play the A/D7/Dm on the fifth fret. But rather than play a Dm shape barred on the 5th fret (tricky!) to get the Gm, they'd play a Em shape on the 3rd fret. Not using a capo gives you the freedom to move your barre around.
But you could also play the song using the standard shapes for those chords; it will sound different from the song with the capo, since these are different voicings of the same chords, but it will harmonise properly. It will be up to you which version you prefer.
Because it contains both an A and an Am this isn't a very common set of chords. However, A, D and E are the 1st, 4th and 5th in the key of A, and I, IV, V is a very common set of chords, so it's worth remembering what they are in various keys:
- C, F, G
- D, G, A
- E, A, B
- F, Bb, C
- G, C, D
- A, D, E
So you can try your song in at least 6 keys without using a capo, for example using Em, E7, Am and B. Some keys will need a barre for certain chords; others won't.
A capo becomes indispensable when you want to transpose chords that can't be played without open strings.