The problem with interpolation is covered in the documentation for  /PATTERN/msixpodualngc in Regexp Quote Like Operators.

If "'" is used as the delimiter, no variable interpolation is done.

This is not enough to solve your problem with the newline because $ is a zero width assertion. Assertions

Besides "^" and "$", Perl defines the following zero-width assertions:

In this case, it is probably best to match the newline explicitly and not use the anchor at all.