Data transferring without iCloud backup

My phone touchscreen is broken and I’m getting a new phone and I would like to backup my broken phone. But I have no current way to back it up since my touchscreen is broken. Once I get my phone will quick start transfer everything without a iCloud backup needed ? If not is there a way I can use my phone screen with a 3rd party device…? More like screen mirroring but I can actually move it.