What you are describing is a stream cipher created from a generic PRNG. However, an actual stream cipher should be used instead. PRNG's are dangerous in the sense that they are sometimes pre-seeded or re-seeded by the system. If there is no mandated algorithm then the underlying implementation may change as well. Otherwise: yes, a stream cipher can of course be used to obtain confidentiality of the messages if used correctly.
AES in CTR (counter) mode is a pretty safe bet.
You seem to be using encrypt-then-sign. Usually sign-then-encrypt is used. The issue with encrypt-then-sign is that somebody can re-sign the ciphertext and claim ownership of it. The other aspect is that after decryption the signature is lost.
Therefore, usually sign-then-encrypt is used for asymmetric primitives such as ECIES and ECDSA. This may not be a problem in your specific protocol however; you could just want to allow only signatures from one specific private / public key and you could be fine with protecting the ciphertext in transit.
Once you have established a key and authenticated the exchange then you could also use AES-GCM or another authenticated cipher to send messages providing integrity, authenticity and of course confidentiality of the messages.
Note that there is a lot to do if you want to create a transport protocol. In place encryption is a bit easier wrt to possible attack vectors. If the system is secure in the end depends if the possible threats to the system are thwarted.
We can therefore not tell you if your approach is safe; we can tell if it is unsafe though :)