Do you need two spaces after a period? A can of worms, I’ll agree, but I’m pretty hardcore when it comes to the Chicago Manual of Style. Also, I’ve worked with editors and publishers out there in the writing world, and guess what? They prefer one space. Make their life easier and use one space if you can.
From the Chicago Manual of Style in answer to a question about one or two spaces after a period:
The view at CMOS is that there is no reason for two spaces after a period in published work. Some people, however—my colleagues included—prefer it, relegating this preference to their personal correspondence and notes. I’ve noticed in old American books printed in the few decades before and after the turn of the last century (ca. 1870–1930 at least) that there seemed to be a trend in publishing to use extra space (sometimes quite a bit of it) after periods. And many people were taught to use that extra space in typing class (I was). But introducing two spaces after the period causes problems: (1) it is inefficient, requiring an extra keystroke for every sentence; (2) even if a program is set to automatically put an extra space after a period, such automation is never foolproof; (3) there is no proof that an extra space actually improves readability—as your comment suggests, it’s probably just a matter of familiarity (Who knows? perhaps it’s actually more efficient to read with less regard for sentences as individual units of thought—many centuries ago, for example in ancient Greece, there were no spaces even between words, and no punctuation); (4) two spaces are harder to control for than one in electronic documents (I find that the earmark of a document that imposes a two-space rule is a smattering of instances of both three spaces and one space after a period, and two spaces in the middle of sentences); and (5) two spaces can cause problems with line breaks in certain programs.
So, in our efficient, modern world, I think there is no room for two spaces after a period. In the opinion of this particular copyeditor, this is a good thing.
If you simply can’t break the habit (which you should try because in the long run it’s more efficient), make sure you reformat your document before submitting it somewhere professional. Look up the manuscript guidelines or style guides for particular publishers and agents, if they have them available.