Should I keep my abs constantly engaged?

No! Do not keep your abs (or any muscles) contantly engaged. Constant core engagement can cause chronic muscle tension in the core and pelvic floor, leading to a host of symptoms that include pelvic pain, pain during sex, tailbone pain and constipation. Instead, focus on maintaining good posture with neutral spine (neither tucking your tailbone nor sticking your butt out) and allowing your core muscles to relax when you are at rest.
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Think about your biceps- you do not strengthen them by keeping them engaged all day with your elbows bent. Instead, you strengthen them during a workout, and use them functionally as needed during the day. When you perform a physical exertion, such as lifting a bag of groceries, exhale and engage your core “up and in” toward the spine as you lift the bag. When you are not exerting yourself, your core muscles can and should relax.