> We've tried everything we can think of.
As stated in all the site guides for StackExchange you should include what you have tried already in your question. Also include the results of what you tried - i.e. any error messages that resulted. Otherwise we are probably missing vital clues that would help us help you, and we may waste time suggesting things you've already tried.
To answer the direct question: it should be OK to restart everything, but it almost certainly won't help you fix the problem.
You certainly shouldn't just move database files around while SQL Server is off, instead either backup the restore to another location (using
WITH MOVE if talking to the DB directly or the relevant file location options in SSMS's restore dialogue) of (faster, using less temporary space, but slightly less safe) detach the a database with
sp_detach_db, move the files, and reattach with
sp_attach_db (or there are options in SSMS for detach and attach, which you may prefer instead of hand-writing calls to those to system procedures).
Before doing anything, including restarting services, make sure you know why the space is suddenly low. If it is obviously due to database files growing carry on, but if there is no obvious use of space like that do check for filesystem corruption just in case, and make sure that is fixed before proceeding.
If the problem is partly due to sudden growth in transaction log files because of a glut of large transactions since, you can probably buy yourself some breathing room back taking backups then truncating the transaction logs. Remember that if you are log shipping or using incremental/differential backups as part of your normal backup regime this will break your backup chain and you'll need to restart it.