What format is the 350mb file? is a text sql file?
Do you have ssh access to log into the server command line?
Putty is the tool to connect via SSH, I am just not familiar with GoDaddy and their setup and you packages you are on.
Also like I say I have a guy who is part of sales and tech for godaddy and runs a large team as well so if you pm me the package you were on and new one I can see what he says.
If attachments/images are saved in the database it makes it easier, if they are stored as files you will need to move those across.
You would be best, potentially to wiped the new database and reimport fresh, it has been a quite a few years since I had a play with SMF though.
For example if you have ssh you can upload the backup sql file using WinSCP then run mysql -u{username} -p -h {SERVER ADDRESS} {database name} < filename.sql
Right now, with the ending of the old server, I would take a fresh backup of the DB and copy all files hosted just in case via FTP/SCP.
If its all a bit too much feel free to drop me a PM with a discord name, or other type of comms and I can live chat about it tomorrow morning, I just have to crack off to bed soon as I have to run the kids in the morning.
EDIT: Ugh... cpanel....
EDIT 2: I have send my mate a message just in case he can put a hold on the old server being end of lifed but he will be in bed, told him I dont know you directly so to prod support etc to speak to you as I know no details, dont want to breach any GDPR issues (he has been told not to inform me anything) but if he can prod from his end it may help and worth a shot.