what your transaction actually showed that sent bitcoin, usually no.
based on my experience, I always make backups private key in a note (usually I use excel).
Uhm, I hope you encrypt that.
I can only assume you:
1. try to find out the private key that is in the bitcoin-qt application, how: go to the debug window type "dumpprivkey [bitcoin address you]" >> notes and confidential
2. try to find out the private key electrum bitcoin you sent earlier, how: go to the console tab >> type "dumpprivkeys([bitcoin address you])". or right-click on bitcoin address and then click the private key >> >> copy
Isnt 2. the same as 1. only for a specific address?
3. then you try to look for a computer with a decent internet speed so that synchronization can run faster (I recommend the VPS / Virtual Private Server)
uhm, why?
4. download / install electrum in VPS >> imported private key in electrum
The way I understand OP they already have a working electrum version, why would they need a different machine for this?
5. wait until synchronization is complete (usually 2 hours)
Did you ever use electrum? I never had it sync longer than a few seconds.
6. after the synchronization is complete send to the destination address (eg exchanger)