My favorite moisturizer is
Nature's Gate Oh What A Night, not exactly drugstore, but they do sell it in Bed Bath and Beyond and online. I've converted several people to it too
Cetaphil Lotion is a favorite of many people with easily irritated skin (it's lighter than the cream) because it's very gentle. I don't hate it, I just don't find it to be better than my other moisturizers so I never bought a full size. It might be a good jumping-off point, and then you can tailor your moisturizer wishes based on what worked/didn't work for you with Cetaphil.
Alba Botanica makes a really nice oil-free facial moisturizer with green tea and aloe that might be comfortable for the Southern humidity! It has a really great texture, I find it to be very similar to that of Embryolisse (beloved moisturizer/primer of MUAs). It takes a few minutes to sink in but is then velvety-matte if you haven't used too much. Strong tropical scent that luckily fades quickly. Also very gentle, and has "natural" ingredients if that's important to you. I think it's around $15 at CVS, but 1) you only need a TINY amount so it lasts forever and 2) it's $10 or something online or at Target.
Biore and
Garnier both make lightweight gel-textured moisturizers for hotter climates. I haven't used them but you could check the reviews on makeupalley.
If your skin can tolerate AHA's,
Alpha Hydrox 10 or 12% cream (or the Walgreens generic dupe) is light enough for summer nights, but still seriously moisturizes and prevents flakies and pimples by speeding up exfoliation. You will need to wear sunscreen the day after you use it, and you should do a patch test first because you might be one of the weirdos like me whose skin gets milia from AHAs!
Also, on nights when my skin feels a little congested, I just moisturize with tea tree (recommended;
Jason Naturals makes a great one you can get at Whole Foods) or aloe gel - they fight spots and are moisturizing enough to be used alone if your skin isn't very dry.
hope that helps a little!