For the first sentence, normally people say "shake off this cold". I've never heard anyone say "I can't seem to shake this cold." For the second sentence, I would also lean to "shake off". But in this case, "shake" is acceptable as well.

Both are fine. We can use the verb "shake" informally to mean get rid of, or put an end to, something. We can also use the phrasal verb "shake off" to successfully deal with, or recover from, something. If I had to choose, I would use "shake" for both your examples.