|It's possible that the diapers may have started to get quite worn and lost some of their absorbency. Over time all fabrics get thinner so they're less absorbent. Honestly though I would think you'd have been able to tell quite easily if they were getting to that point.
Yes, you do need to use a cover all the time. When baby pees, the moisture will seep to the outside of the diaper and get clothing/bedding wet. The cover is the barrier that prevents that from happening. You'd need a ridiculous amount of absorbency to prevent the outside of a diaper from getting at least pretty humid so if this happens it's not because the diaper isn't absorbent enough. Some do go coverless but when they do it's in specific situations only (ie at home, without pants on etc...) and it's often with older toddlers that are walking around much of the time so you don't have to worry too much about a small baby that's always laying on something or in someone's arms.
Most people don't need liners during the day but usually need at least one or two for nighttime. Some very heavy wetting babies using the One-Size do need one during the day, you'll know if this is the case for you if the diaper gets saturated (ie totally wet all over) even if you change frequently enough (depending on the age of the baby that should be every 1.5-3 hours). This isn't as frequent with the Sandy's as they are a little more absorbent.