LaTeX breaking up too many words

I've always just used \raggedright for this, but I've never done that for an entire paper. This FAQ (which is one of my go-to LaTeX resources) recommends the ragged2e package.

Oh, and a side note: I never pay attention to bad boxes unless I can see them in the output. This is probably dumb, but LaTeX is much pickier than I am.

While, yes, you could make TeX less picky or even go the ragged option, this is best fixed by the microtype package. Just loading it should do the trick:

\usepackage{microtype}

It provides two features to improve the typesetting of paragraphs:

• Margin kerning: characters, especially punctuation, will protrude a small fraction into the margin. Practically, this has the effect of very slightly increasing the line length and (IMO) visually looks very nice.
• Font expansion: as well as stretching/shrinking the space between words to create a flush paragraph, the characters themselves are stretched/shrunk by very small amounts (less than one percent). This is visually imperceptible at the character level but, surprisingly, makes a huge difference at the paragraph level.

you should set a hyphenation penalty somewhere in your preamble:

\hyphenpenalty=750

The value of 750 suited my needs for a two column layout on letter paper (8.5x11 in) with a 12 pt font. Adjust the value to suit your needs. The higher the number, the less hyphenation will occur. You may also want to have a look at the hyphenatpackage, it provides a bit more than just hyphenation penalty.