I, too, am experiencing this. We're into the third week of school, and every day is a struggle. The principal has suggested that I no longer walk her to her class. I meet the principal, counselor, or another teacher at the entrance of the school. We make the separation there. No long goodbyes. A quick hug and kiss, and then I have to be firm. Yes, sometimes she is holding on for dear life and has to be lovingly pulled away from me. It's hard for me to see that, but everyone reassures me that she is going to get over this before too long.
Try to be patient and supportive and, as much as you can, take this all in a matter-of-fact way. The start of school, even with children who fundamenatlly are doing OK and who do not display separation anxiety in other settings, can be stressful. The good news is that this reaction will go away, and you don't really have to adopt any special intervention. Nothing bad is happening and things will be OK. There are many parents in the same 'boat' as you during this time of year.