I use KeePass, open source and free. I’ve used it for years without any problems.
It can generate passwords based on the parameters you choose. Your passwords are organized by financial, medical and website groups, etc.
One password to rule them all - just don’t forget your master password.
You can keep your passwords in the cloud, but I wouldn't want my passwords there.
Re: LastPass and some other password managers: You have to log in to LastPass which means they have your passwords on their servers. Again, I wouldn't want my passwords on someone else's server. If LastPass goes down, you won't have access to your passwords until their service is restored. It does happen and, according to Murphy's Law, always at the wrong time:
Some people like this option because it's convenient and there is a free version, but it's not for me.
You can keep your password manager files on a removable USB flash drive. This solves your problem of not wanting the passwords on your computer, but the database is more easily corrupted. Flash drives are less reliable than a hard drive or SSD drive and the flash drive has to be properly ejected from the USB port before removal or data corruption can occur.
You can keep your passwords on a tablet or smart phone, but these can be lost or broken. And you will need to be able to backup your data which isn't always simple on a mobile device.
I have KeePass on my home personal computer and back up the KeePass folder to a flash drive. It's simple and easy. If you store your passwords on your PC, you will need to keep your anti-virus up to date.
Wherever you decide to keep your passwords, routinely make a backup!!