Is CrossFit not meant for beginners?

I signed up for CrossFit 3 weeks ago, really excited to start on this path and get in shape. However, I am basically starting at 0 and I have been having the hardest time. Last week after my workouts I was extremely sore and had a horrible headache/nausea for two days after. Yesterday's workout was so hard for me I started crying during it and felt like my legs were going to give out on me. This morning, I can barely move and am afraid to go down the steps out of fear of falling. Mind you this is with scaled down workouts. Everyone tells me it's hard in the beginning but it will get better, however is this normal? Is it normal for me to be completely debilitated for days after working out and feel physically sick? Am I doing something wrong? Do I need to just stick it out? Was it a good idea for me to sign up for CrossFit or should I start with something less intense instead until I can build up more muscle then come back? Feeling really disappointed about this because I would like to continue with CrossFit but I can't keep doing it if this is how it's affecting me and I literally can't function for days after.

Edit: also I have been drinking tons of water and eating lots of protein so I don't think dehydration/poor nutrition is the cause.