A fast lens is the better first upgrade.

Your main problem is subject motion blur, so you need a faster shutter speed. A wider-aperture lens like a 35mm or 50mm f/1.8/f1.4 gives you more light, letting you raise shutter speed without pushing ISO even harder. That’s usually the most practical solution for indoor events.

Flash can help, but it’s often restricted at events, can distract people, and may produce a harsh look if used on-camera without good technique. Event shooters do use flash, but often off-camera or as subtle fill, with permission and more setup.

A 35mm prime is often especially useful for events: it’s easier to handhold, gives a wider view, and can feel more natural in tight spaces. A 50mm can also work if you want a tighter field of view.

If you do use flash, try to keep ambient light in the exposure and use flash only as fill when possible.

Also, in low light and at wide apertures, autofocus becomes more critical. Practice your AF settings and shoot plenty of frames—event photography often rewards volume because timing, focus, and composition all have to come together at once.