Do I need a separate sunscreen, or is moisturizer with SPF enough?

You need a separate sunscreen. Most people don’t apply enough moisturizer to get the labeled SPF protection — you need a nickel-sized blob for your face and a shot glass worth for your body, which is way more than anyone uses. Moisturizer SPF is usually lower (like SPF 15 or 20) and often not broad spectrum, while you want SPF 30+ covering UVA and UVB. Plus, reapplying is awkward with moisturizer. Just buy a separate sunscreen you actually like wearing.