I agree with Teacher Erin. In addition, in the US, we would be unlikely to SPEAK these sentences without the word "that". It sounds awkward without "that" and native speakers usually go for the best sounding way to say something. :) We have all done things in our lives that we were ashamed of. We have all done things in our lives that we are ashamed of. We have all done things that we weren't proud of.

These are great sentences and in general they are correct. I will say one thing... In the examples, you use the past simple (weren't proud of/were ashamed of/regretted). If now you are still not proud/are ashamed/regret, then use the present simple to show you continue to feel this way. Examples: We have all done things we weren't proud of (in the past, we were not proud) We have all done things we aren't proud of (and we continue today to not be proud)