Do I need to use AE/AEAD in my particular scenario? Or is it redundant?
In general signature creation contains the hashing part within the algorithm. A signature algorithm may also contain a padding mechanism such as PKCS#1 v1.5 or PSS for RSA. Finally it contains a one-way trap door function (modular exponentiation within RSA). Encryption has other requirements, and uses a different padding mechanism.

Basically you are encrypting a plaintext, and signing all the data including IV within the message. I don't see the need for either a MAC or an authenticated cipher; the signature is perfectly fine as authentication tag. That does not mean that your current scheme does not have weaknesses:


it does not provide forward secrecy (i.e. if your secret key is compromised, all the messages can be decrypted)
you have to keep both a private key and share a secret key
asymmetric operations are expensive (with regards to CPU) and you need a private and public key operation per message
no signer anonymity (eavesdroppers can prove that messages were sent by a specific sender)
as others have pointed out, encrypt-then-sign may have vulnerabilities you should be aware of in your protocol


Those drawbacks may be resolved by a careful choice of a TLS cipher suite (if you just require transport security).

Finally you may want to think about how you are going to setup key management.