I think it's best to go straight to 5.4. Which things to look into, please see the migration document in the docs:
https://www.llblgen.com/Documentation/5.5/LLBLGen%20Pro%20RTF/migratingcode.htm#migrating-generated-code-from-v3.5-to-v4.0
(that's from 3.5 to 4.0, then work your way upwards).
You'll not run into most things, however you will run into the move of the csproj files in 5.4 (if you're using adapter) so read that first (it's at the top of the document I linked above).
Use source control (or a copy of the generated source and your code) and generate code on top of the old one (after moving the csproj files if you're using adapter), reference the new runtimes (preferable from nuget) and try a compile. Make sure all files are overwritten. WIth the migration document check whether the issues you run into are documented there and what to do if you run into them.
If you don't want to make the plunge directly to 5.x, you can always go to 4.2 or 4.1 (though you still need to upgrade to 5.x if you want to receive bugfixes). If you have a valid v5 license attached to your account you can download any version below it when logged into the website and go to 'My Account' (it'll be added once logged in) -> downloads -> version. If you need a v4 license to use with 4.2, please let us know and we'll mail you a v4 license so you can use e.g. 4.2 instead
If I may: what issues did you run into exactly? Strange compile errors? Or breaking tests?