Your little girl has had some big changes recently. Other replies have covered this and they're right on.
a couple ideas: Call your daughters former caregivers, the lead teacher/caregiver she was with, if possible. Ask her if you could meet for coffee and if she could tell you what *they* were doing when tantrums and the acting up started, and how they dealt with it. Sometimes, having the continuity helps. Pay her if you need to (offer her a $25 consulting fee or something... it may be worth it.)
Second, for toddlers, I love this book :"The Science of Parenting " by Margo Sunderland. Despite the title, it's actually a pretty fun, easy read and will help you know what to expect of your daughter emotionally and how to 'meet her' in this way. Toddlers are all about the social and emotional development.
16 months may be too young for Time Outs to make a positive impact. Dr. Sears doesn't recommend them until at least 18 months. I didn't use them with my son until he was well into three. They didn't make sense for us and distracted us from getting done what we needed to do. Each child is different. I used a lot of positive guidance at this age.
Kids will divert their eyes from intense adult gaze at nearly all ages. Esp. during discipline. It is an intense experience for them. Try talking to her body parts. If her hands are grabbing: "Can I talk to your hands? Hands, I need you to let go now while I change the diaper. You can hold this (offer a toy) while we do it. Can you shake the toy? Can you show me your fingers? Can you wiggle them?" Distraction, distraction.
Less talking, more physical assisting. Kids need less talk and very simple directions. Don't worry about explanations so much at this age. Give her the direction you need to give, model it if necessary, and then help her do it after giving it once or so.
My last suggestion is to consider if you *might* have a spirited child. NOT the euphemistic bratty 'spirited child', but a child who has strong opinions and particular feelings about what needs to be done and how. Playful parenting books might help, but do also look into some materials on truly spirited kids. I've worked with several of them. They can really give us a run for our money as toddlers and preschoolers, but the ones I know have become very self-motivated, neat, exciting young people who have done well academically and are succeeding in their lives. They have a very strong sense of what is 'right' to them. I'm not saying that your daughter IS spirited, but it might be worth checking out to see if your daughter fits into this particular category and to gain some strategies for working with her personality instead of against it.
You sound like a really nice mom, not a crappy parent at all. :) We've all been there... just ask me about the time my son had a tantrum and fell into a metal stand at the grocery store and gave himself a black eye! THAT was a day!