How to move to another XMPP server

I have an XMPP (Jabber) account on a public server that is not reliably reachable any more (looks like the server is not really maintained any more). Therefore I want to migrate to another server. I already have registered an account on another server; but how do I move my contacts there?

Maybe this is a "pointless" question; but my understanding is that other people know that I contact them via the account on the old server; and if I use the new account, wouldn't they need to accept the contact request first? Also, how do I tell my XMPP app to contact them via the new account at all?

I'm using the Conversation app under Android.

Maybe this is a "pointless" question; but my understanding is that other people know that I contact them via the account on the old server; and if I use the new account, wouldn't they need to accept the contact request first?

They would, yes. XMPP does not have any hidden long-term user identity that would let others associate your previous address with the new one – other than the user@domain itself. So if your new account is at a different domain, then it's a new identity.

(Yes, there's OMEMO, but that's built as a separate layer on top of XMPP and doesn't interact with the underlying system of accounts and subscriptions; it just exchanges generic "events".)

In that regard it's no different from email, or even ActivityPub "fediverse" (the Mastodon/Misskey/etc. network, which does have cryptographically bound identities and a half-baked migration system, but migrating to a new account still involves requesting new 'follows' in the outbound direction).

Also, how do I tell my XMPP app to contact them via the new account at all?

Depends entirely on the XMPP app. It ought to have a feature to export the "roster" or "buddy list" or similar, and to import them back (to the new server).

Since Conversations doesn't have that feature, set up both accounts on a different client such as Gajim which has "Preferences > [account] > Contact List > Export". The roster is saved server-side so it should automatically show up in Conversations once imported.