Uhhh ... mind disclosing what issues you are having with current?
Yes. Possible .... but have to have command line on both and using moosh (a command line only tool) on both.
What version of Moodle to what version of Moodle?
Briefly (not really) ...
You have to install/check all the same addon/plugins in the new.
Themes are plugins ... probably best to force both sites to use 'boost' until every course is migrated. Plugins that don't have a compat version on new server causes problems with restores if courses used those plugins.
New server has same categories - done via Moodle Admin GUI so you know the category ID numbers (moosh restore uses category ID's).
On old, do a query to find all the course ID's in a category.
Then using moosh command with a bash shell script looping through all the courses in a category ID saved to a directory just for those courses - like 'ss' for all social studies courses.
XFer the ss directory to new in a file system repo - so you can see them via GUI as well as from command line. Then use a similar bash shell command to loop through the backup files using moosh command to restore each backup to designated category ID.
Repeat with another category ID only different directory to which to save.
Potential gotcha's - admin user on both systems are the same and/or admin user has been enrolled in a course.
Space - on both servers - making massive full user backups can require more than you have!
If linux might have to run such commands with a 'nohup' (no hang up) in the background for large courses.
PHP and DB tweaks on both servers time for a script to run, memory a script can consume, max_input_vars, etc. will ensure things work.
Its doable ... but might take all day long ... one is almost 'brain dead' when finished. :| Once did a site like that for over 4000 courses in something like 20+ categories - 3 pots of coffee - in about 8 hours!
After looking at all that, might encourage you to address the issues you are having and then do a site migration instead! So we come back to the first question: "... disclosing what issues you are having with current".
Just looked back at your older postings. So if attempting to go from Amazon hosted Windows to Amazon Linux there is a learning curve - plus not sure one can get moosh installed and running on Windows! Guess you haven't resolved those issues!!!
'SoS', Ken