As @kelalaka already mentioned, standardization is not finished yet. There are candidates for PQ-crypto, but they are still analyzed and revisited. Here you can find the current finalists and alternatives. (Note: Sike is not considered an alternative by most cryptographers I know, because of this paper).
But the main reason I make this answer: I think you got a misconception. In general you don't want to use asymmetric cryptography for encryption. The reason for this is, that is not efficient. Symmetric applications (like AES) are more suitable for encryption.
Therefore most applications use hybrid encryption schemes. Simply said, asymmetric crypto is used for the key exchange and then symmetric crypto is used to encrypt the data. This is more efficient, especially if you want to handle large amounts of data.
The good thing here is, that symmetric crypto can be assumed to be PQ-secure to some extend. Although there is a PQ-attack against symmetric crypto, the Grover algorithm, symmetric crypto is not broken in the same way as asymmetric crypto is. Grover's algorithm can be seen as an attack, with runtime $\mathcal O(\sqrt{n})$, where $n$ is the security parameter. For your application (AES-256) that means, if someone could run Grover's algorithm, your assumed bit security drops from ~256 to ~128. This is still assumed to be secure for normal people.
From your description, I don't see a problem with only using AES and already being "PQ-secure" to some extent. I see more of a problem in finding a (commercial and untrusted) server that already uses PQ algorithms today.
