You don't need to re-sign anything, you just need your watching wallet on the Online PC to be synced up (ie. connected and showing a green dot)... Once it is connected properly, it'll find all the transactions and update the status of these transactions accordingly
As for how to proceed, you need to decide which type of install you want... do you want to have it "installed" or continue to run from the "portable" exe... Then just set it up accordingly and try to stick to one version.
For "Installed", Simply run the 3.0.5 installer, move your old wallet into the new wallet file directory and you should be good to go
For "Portable", Simply put the 3.0.5 portable .exe into the same place as the old portable .exe and you should be good to go
NOTE: in either instance, you should make sure that you have checked the digital signatures of the version of Electrum you have downloaded.
I am going to install the 3.0.5 Installer and move the old wallet here. This will be on the off-line. I assume I have to upgrade to the same version the on-line pc as well. If yes, will I have to use the wizard and the public key to create my watch-only ?
In any case will go back again to Abdussamad guide that I already bookmarked some time ago and that I think it's really well written and explained.
It's ok (in the sense I guess it's important) for the digital signature check (just hope it's not another 'burden' to figure out
Thank you very much for your help and understanding (I reckon it might be quite annoying ....with dummies!)