You are actually quite correct in your assumption that there needs to be a multiple upgrade process if you want to take your course into the new Moodle. Go from 1.9.3 to v1.9.19+ first. This will add changes to your v1.9.3 database that make it easier to make the next step up.
Then upgrade that to Moodle 2.2.11 This is the last legacy version of Moodle where you can make the transition with the courses and users database. It leaves it all intact while changing the database to accept the changes made to v2.x.x Moodles.
Then upgrade to a more recent version such as 2.6.1 This is a mid-point, if you like, from v2.2.11 to your destination.
Then upgrade to v3.x. The database should remain intact through this rather large change.
Caution, or advice, or maybe teaching Grandma how to suck eggs, whichever way you see it... I seriously suggest you leave your current production Moodle where it is. Create a new server if you have to, or a new site on a different server or a new site on the same server, or anything that allows you to do the upgrade without really affecting your current production site. Copy your production site to the new site, making sure that all the addresses and everything are pointing to the new site, check your config.php file closely. Do your first upgrade, then test. If all is working, do the next upgrade and test. This is where it is most likely to fall over, which is why you want to not be working on your production site. Test thoroughly, everything you can think of, then assume you have missed something. Then upgrade and test until you get where you want to be.
By this time, you are likely to have run into the issues of PHP not being the right version. The v5.x PHP will, AFAIK, not be compatible with v3.x of Moodle, you may need to upgrade the PHP to v7.x... So expect a bit of trouble along the way.
As you are no doubt aware, this kind of thing causes much grief and angst and general mischief for any admin, so be patient and just take your time. Then you will have to look at your upgrade policy and accept that it is just not working for you. I only update once per year and that is sufficient for my client. It also means not too much work or other issues for me. Seriously suggest you think about it and not leave it for so long.
(And all this is from a real fan of v1.9.xx still grieving about missing all those little idiosyncrasies that made Moodle such a joy to use... )
Your only viable alternative is to start from scratch... Cheers.