The Matrix network is the closest you're likely to get to Discord's features.

Nheko is a Matrix client that I believe can do screen sharing.

Eventually, whatever Matrix clients support Element Call might be what you want, but it's in beta for now.

Jitsi Meet might also be worth a look, although its (optional) end-to-end encryption was too demanding for some laptops last time I tried it.

All I really need is the ability to do voice calls and screen shares.

Have you looked into Jitsi?

I would recommend using Matrix. But if you really want the "Discord experience" than I would suggest Revolt.

Element for Matrix

Since everyone is only recommending self hostable options and no hoster, here is a new candidate:

OpenTalk on tchncs.de

https://talk.tchncs.de/

It is also free and open source and the server is hosted by a german who takes donations.

The also have a mumble (like Teamspeak but as you said without screenshare) and matrix instance running, which are great but not for your needs.

Jitsi, for sure

Signal. Works both on mobile and linked to desktop.

It should also work on Element Web, but not desktop clients.

Matrix using element for the client and element call or jitsi should do the trick