Definitely look for a new job.
Years ago I was in a similar position - had to beg for work and just felt like I was being paid to do nothing. It was so demeaning.
On top of this, the person who was supposed to be training me, started to exclude me - she started to go around the cubicles and ask people if they wanted her to hand in their time sheets. She ignored me.
She would go around asking everyone if they wanted to go to the canteen for morning tea, and ignore me - fortunately there was a nice young lady who would then ask me to come along.
I kept asking her for work, and she said that it would be too hard for me (it was basic financial data in excel! Plus I have a double degree!)
About 4 weeks in, she told me she had previously resigned from her role, and they specifically brought her back in to train me, but she decided I couldn't follow it (she knew nothing about my skills!) and told me it was just easier for her to do the work herself.
When I was given work, I suggested a very simple improvement on one of her processes - she said it wouldn't work and I didn't know anything. Fortunately she was away the next week, so I had to do the task on my own and I changed how it was done to my suggestion. When she returned, she gave me the task and I followed my new process and was finished fairly quickly - I had turned a 3 hour daily job that took 2 people, into a 15 minute job for one person. She didn't believe I had done it correctly and she spent the next few hours checking the paper work with someone else.
Happened again the next 2 days. I was able more efficient and had reduced the pile of paperwork significantly, and saving over 5.5hours of paid work. She then asked me about my process and she started using it but she never gave me credit for it.
I left not long after because I'd had enough of twiddling my thumbs and feeling excluded.