Electrical Outlets - Three Pronged No Ground Wire

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Can anyone recommend that safest yet least costly way to deal with the following issue.

It is past time to replace my electrical outlets (1967 condo). I changed one outlet and everything went as planned: neutral, hot, and ground wires. Then, when I got to the outlet right next to that one, it had no ground wire, yet a three pronged cover. I've read not to put three pronged outlets in areas where there is a missing ground wire.

When I plug in the three pronged tester is says "correct", but there is no ground wire on some of these "correct" outlets.

Are there any other options besides trying to have the HOA rewire my place with a ground wire everywhere. They'll probably hire a bad contractor causing this place even more danger than it could be in now. The HOA here likes to cut corners and it often makes things worse.