Yes, if the multiple stems come from stem bulblets, they should be much more easily removed without disturbing the mother bulb.
But it sounds much more likely that the mother bulb has divided, as Australis shows above. A profusion of divisions, like yours, is often caused by hot conditions (mostly in the soil), the bulb planted too shallowly, or physical damage to the bulb (but not life threatening). Sometimes it is just plain vigor, and with good soil like yours, that very well could be!
You have such good, friable soil that I wouldn't hesitate about digging it up at the right time of the year. The biggest set back that happens when digging and planting/transplanting bulbs is loss of the roots. In most people's gardens, their soil is much heavier, and most of the root structure tends to break off. We don't have to worry about the tiniest roots; they grow back very quickly. We want to keep all the larger roots, so they can be repositioned optimally when transplanting so that there will be ample space for smaller, feeder roots to grow. This shouldn't be any problem for you in your loose soil. Using a digging fork (not a shovel or pitch fork) will more easily accomplish the task. If you don't have one, I recommend you invest in one. You may find it preferable for many chores.
