They can be used but I don't think they are "colocations" in U.S. English. Much more common would be:
...he sincerely apologized...
...he made a heartfelt apology...
If "apologized" is not strong enough, "sincerely apologized" is the best choice.
"Solemn" collocates with "vow" or "promise" or "pledge."
"He made a sincere apology, and gave a solemn pledge not to do it again."