Can I overlock without cutting the fabric?

Yes, and there are two ways.

Disengage the blade. Most overlockers have a lever or knob that retracts the upper blade. Check the manual, because the mechanism varies and forcing it is a good way to knock the blade out of alignment.

Or guide the fabric to the left of the blade so it simply passes clear. This works but requires attention, and a moment of inattention costs you the edge.

When you want this:

• An edge already cut to size, where you only want the finish. Trimming would make the piece smaller than the pattern.
• Curves, where the blade tends to shave a little more off the inside of the curve than you intended.
• Applying elastic or binding, where the edge is being covered anyway.
• Anywhere a mistake is expensive — the blade is unforgiving and there is no undo.

When you should let it cut. Ordinary seam construction, where a freshly trimmed edge is exactly what makes an overlocked seam neat. The blade produces a clean, consistent edge for the loopers to wrap, and disengaging it on a hand-cut edge often gives a wavier finish.

One caution: with the blade disengaged, it is easy to feed too much fabric under the foot and end up overlocking a fold rather than an edge.