The quotes seem to be fine at my end...at least, warnings doesn't indicate any special issue with them. Warnings just says there's an unrecognized escape in the line where I escaped the period and indicated a space with \s. As far as my eye can see, there should be no error there.

Line under discussion:
$query = "(St\.\s)(Mt\.\s)(?=Helens)";
Unrecognized escape \s passed through at blah... line x

Perl is saying that it figures you made a mistake with \s. It translated that into a single "s" character. It also translated \. into a literal single character of '.' but it knew about escaping a period and Perl didn't complain about that.
Consider the following:

#FROM INCOMING FORM INPUTS
$query = '(St\.\s)(Mt\.\s)(?=Helens)';  #right way
print "$query\n"; ##(St\.\s)(Mt\.\s)(?=Helens)

$query = "(St\.\s)(Mt\.\s)(?=Helens)";  #your way
print "",$query,"\n"; ##(St.s)(Mt.s)(?=Helens)
print "$query\n";     ## same thing (St.s)(Mt.s)(?=Helens)

$query = "(St\\.\\s)(Mt\\.\\s)(?=Helens)"; #ok, but confusing
print "$query\n";    ## (St\.\s)(Mt\.\s)(?=Helens)
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Fixing the quoting has real consequences in terms of what $query winds up being!
I always "use warnings;". I very rarely ignore a warning, with the possible exception of working with old code and the "deprecated syntax" warning. However, in all cases I do strive to understand what the heck is wrong that Perl is complaining about and then try to "make Perl happy". Sometimes with deprecated syntax, the error may be so pervasive that is not practical.

I understand that in your production code, this string will come from elsewhere instead of an assignment statement like above. Be that as it may, I still strongly advise understanding what a Perl warning is telling you and fixing all test code so that it runs without any warnings. I have heard that Perl runs slightly slower with warnings enabled. I have never benchmarked that because this just hasn't been a significant factor in my work. I recommend leaving warnings enabled at all times.