Your 1st, 2nd and 4th images are camera shake blur.
The 3rd photo might be soft due to shallow depth of field and the point of focus... Of the three subjects in that shot, the two on either side seem relatively sharp and are slightly closer to you. The subject in the center appears slightly farther away and not quite focused upon, with that combination of distance and an f/2.8 aperture.
I agree... you could have bumped up your ISO and that in turn would have allowed you to use a higher shutter speed that would better freeze both subject movement and your own movement/camera shake. 1/15 and 1/30 used in several of the shots is marginal when using a 35mm lens. Image stabilization can help with camera shake, but still requires good technique. But image stabilization cannot do anything to help with subject movement, if any of that's occurring.
Besides that, there are a number of techniques you can learn to help increase the percentage of sharp shots you get at slower shutter speeds.
The most obvious and easiest in this type of situation is to use a flash. The short duration of a flash acts like a fast shutter speed, regardless of the camera's actual shutter speed. Most flashes give the effect of around 1/720 shutter speed.... far better freezing movement than 1/15 or 1/30 second! One problem with flash occurs when trying to balance it with ambient light, to act as fill, at slower shutter speeds. That can cause "ghosting" effects.
Flash also can make for ugly shadows and redeye problems. The trick to reducing that is to get the flash out of the camera's hot shoe (aAVOID built in flashes... they really suck for a number of reasons). Put it on a flash bracket that positions it off to one side and a little high up. If not a radio controlled flash, this requires an off-camera shoe cord. Positioning the flash that way greatly reduces redeye issues and helps move shadows down and behind subjects where they are much less of a problem. So long as the flash is powerful enough, diffusing it in some way also can be very helpful making for nicer images. Whoever gave you advice was pretty much steering you in exactly the wrong direction.
Using a flash to add light to the subject and scene also allows you to stop the lens down a bit, for greater depth of field that would help with problems such as in the 3rd image (re-positioning the subjects so that they are all the exact same distance from the camera is another solution for that).
Yes, a tripod or monopod would have helped with camera shake, but not with shallow depth of field issues or with subject movement. It also would be more cumbersome shooting events such as this, making shots less candid and more formally posed. Most event photographers would shoot this sort of event with an accessory flash, handheld.
If forced to use slow shutter speeds handheld, yes, as others have suggested, your first concern is to stabilize yourself as best possible. Lean against something, work on your stance and how you hold your camera, and don't "stab" at the shutter release button... practice "pressing" it instead. All can help. Also set your camera to a continuous shooting mode and take a series of shots instead of just a single snap. That will increase the odds that one or more will be good, steady shots. In other words, expect a high percentage of shots to show camera shake and to go in the trash... so take extra shots.
The photo below was shot with a 50mm lens at f/2 on an APS-C camera... at 1/30 shutter speed, handheld. No image stabilization, either. In this case, the trick I used to prevent camera shake and get a sharp shot was resting my elbows on a tabletop, sort of like making myself into a tripod...
Here's another example, this time at a much higher shutter speed that prevented any sort of camera shake or subject movement blur... But in very low light where an 85mm lens set to a large f/2.0 aperture rendered shallow depth of field.... In this case I was very careful to focus the person's face and torso (the most important part of the image) sharply, while allowing his hand and the bottle he's holding to go soft due to the shallow DoF:
Flash used correctly can greatly improve images. It's just a matter of using some tricks such as those described above. In fact, I use flash a lot on bright, sunny days... as "fill light" to open up heavy shadows:
I almost NEVER bounce flash off ceilings or walls. Bouncing often causes more problems than it solves. For one, the tonality of the bounce surface has a huge effect on the color of the light. It can be complementary... but far more often it's ugly! Bounce panels I use in studio (a far more controlled situation) are pure white, silver, gold and a mix of silver & gold. Which I use depend upon the subject's skin tones and the color of clothing, background, etc.
Bounce also wastes an awful lot of light... forcing any flash to fire much more fully, slowing down recycling and draining batteries a lot faster. When you bounce flash the light has to travel a lot farther and some of it's absorbed by the bounce surface. Using ceilings and walls, the distances are highly variable and largely out of your control, too. I do bounce large studio strobes out of umbrellas at times, for the especially soft, somewhat "wrap-around" type of lighting that produces. I've sometimes also used a "bounce card" mounted on a portable flash on my camera. But in both these cases the distances and color of the bounce surface are very much under my control.
Built-in flashes simply suck. They're weak & wimpy so don't have much "reach", are located in the worst possible place for redeye and ugly shadow effects, slow to recycle and draw heavily on the camera's batteries to greatly reduce the number of shots per charge you'll get. I highly recommend getting an accessory flash and learning to use it. A built-in flash might come in handy for an emergency situation... or might be used to control an off-camera flash... but in general I recommend avoiding them. An accessory flash is much more controllable, far more powerful and can be positioned much better. Plus it has it's own power supply to give faster recycling and not unnecessarily drain the camera's main power source.