Yes you can use KeePass more securely than how you are currently and you do not need to switch to another password manager. KeePass2 contains a feature called Auto-Type Obfuscation. Instead of copy-pasting the password you can right-click on the password and select 'Auto-Type' and it will then ALT-TAB (switch) to the last window that had focus and start typing in the username and password (or you can configure it to type other data too if you have more or less fields).
To enable obfuscation on the auto-type right-click on the password record and edit/view the record. Click the Auto-Type tab and at the bottom there will be a textbox that says 'Two-channel auto-type obfuscation'.
When this is enabled the username and password details are obfuscated by jumbling it up and entering it out of order and it is also split into parts with some parts in the clipboard and some parts being entered as a keyboard. This way it would be extremely difficult for a keylogger or clipboard catcher to get the whole password or have any idea how to reconstruct it in full.