I.G.
Oh, I have so been in your shoes - and it's not over yet...
It was like someone had flipped a switch on my baby's first B-day and turned her into a toddler, with the tantrums and everything that comes with it.
I just try to take it in stride. We also deal with the clinginess, but I just simply wear her in my ergo if she wants to be held - that frees both of my hands to do what I need to get done. That said, she is in daycare full time, so the little time we spend together I really don't want to deny her that closeness.
Now at almost 20 months I have also started to explain to her if I just need to put her down for a short moment - and ususally will be fine. Sometimes I give her something to occupy herself as well.
In a full-blown meltdown I either hold her, or if she does not want to be held, I sit her down by herself, explain that she needs to calm down before I will pay attention to her again.
I do NOT leave the room and leave her to "cry it out" (we have never done that) but I will also not go back to her until she calms down. After she is calm I will go and hug her, and tell her that it's ok to be upset, but that right now she can't have what she wanted and we will do something else instead (or we do it tomorrow - whatever applies)
At this point these strategies are working very well, and she will ususally calm down within a few minutes. I do not expect the tantrums to ease for at least another year or so, after all it is just a normal phase of their development.
As I think everyone, I do lose my cool every once in a while, especially when I need to finish something or when I am tired from work, and speak to her louder or stronger than I intended. It happens... but I really try to remind myself when encountering these situations that A. she does not do this to annoy me, but because she hasn't learned yet how to handle her feelings - and B. that I am the adult and there is no reason for ME to be upset too.
Try to consciously enjoy the beautiful moments at this age and try to remember them during a tantrum.
Good luck!