Caulk the joints and leave it like it is. The square corner looks really bad. Once you caulk the joints where the small piece meets the larger one, then, fill in the nail holes, sand well and primer and paint the rest of it. Do not caulk to floor.

You can get radius corners for baseboard:

Just google it and many options come up.

Either go get radius trim pieces that maybe will match yor baseboards or just fill and paint what you have IMO it will look fine . IMO it looks intentional with the squared off baseboard I do not think a rounded one would be better.

After putting in the angled trim on round corners like your first pic and deciding that it didn’t look good, I replaced the corners with prefab round ones. Using a multitool the baseboard edges could be squared off without removing trim and it looks better. Installing baseboard like you would on square corners doesn't look as good IMHO but you may like it and it is easier to install.