Is the sentence "I now feel better" correct? Shouldn't it be "Now I feel better"? Or are both of them correct?
Both of them are correct.
'now' is an adverb, and adverbs, as a general rule, can be placed anywhere in the sentence as long as they don't interrupt constituents. This, however, doesn't mean that all the possible placements would be common.
So all of these are acceptable:
Now I feel better.
I now feel better.
I feel better now.
However, this is non-standard at best:
%I feel now better.