Thanks for hosting provider info! Is NS where you are moving to or is that were your moodle is currently hosted?
According to:
https://www.networksolutions.com/web-hosting/hosting-package-comparison.jsp
No ssh (putty) only FTP (hope your new home will provide more [or Network Solutions hosting where you intend to move?]).
Is it a dedicated host or a shared host ... jumping from one shared hosting to another shared hosting may not resolve potential issues - remember it's not just the versions required for MySQL and PHP, but the limits to resources each user/account/site can use.
Best at NS might be what they call a VPS:
https://www.networksolutions.com/vps/index.jsp
Before we go further ... am I just not understanding or are you switching gears here? You are trying to upgrade your current 1.9.5 to the highest 1.9.19 correct?
Can you, archive the data directory, moodle code directory of your current site and download for backup? Can you acquire an SQL dump of the current database?
'spirit of sharing', Ken
PS: might want to check out NS forums: