I'm sorry you're feeing so deflated at work OP. There are ways to deal with this though that may help in the interim while you're looking for another job - and which may also help in future jobs.
From what you're saying, my understanding is that a) you are in a team and everyone picks up jobs/tasks from the same pot of work (I'm guessing customer orders/requests/issues/enquiries of some sort - just from your posts).
You say you work from home and are having systems errors which are slowing down your productivity. It seems that others MAY have a similar issue or may not - but you don't know for sure. I'm therefore guessing what is happening is that the system is slowing you down therefore from a stats perspective it looks like other team members are picking up jobs/tasks at a faster rate - so it looks like they are more productive.
I am assuming you have reported the systems errors/problems to your IT department in all this so you should have a record of this- emails to your IT dept or whatever. The IT problems you are having may therefore be related to your particular system (at my company some people's laptops are better than others!)... or it may be your internet/wifi is causing drop outs - which happens when people are working from home.
Assuming the above is correct, what I would do is this:
- Gather up all your communication with IT - all the times you have reported the tech issues and their responses (assuming they are trying to help you sort it).
- Gather up all your evidence - screen grabs or whatever... showing how frequently this is happening and how much it slows you down.
- Arrange a meeting with your manager and present your evidence - not in a confrontational way - but in a "I'm wanting to resolve this" way. You can then say "this is what is happening - I acknowledge that it is impacting how much work I am able to pick up and despite my best efforts - it is still a problem..." - ask the manager to liaise with IT to help you sort it. If it's an internet/wi-fi issue at home then you need to have a conversation about how they want you to deal with that - if you are working from home due to Covid (ie wouldn't normally) then they need to factor that in.
- In terms of the situation when work runs out - you say "when all the tasks are complete and there is no work to pick up - I have been doing x,y,z to keep busy and be productive - are you happy with this or do you want me to do something else?" - put it on your manager - she then can't complain in future if you are doing what she wants.
It sounds like she's not getting the systems issues which is why she had you shadow another colleague - perhaps she thought they might approach the work differently - or it might help identify if it's a training issue or a systems issue. You need to demonstrate clearly a) what the problem is and b) what you are doing to resolve it.
If it's an IT issue beyond your control and you have done everything reasonable to solve the problem then present the evidence to your manager and ask her to liaise to help get it sorted (my managers are all keen to do this if they can see you are doing all you can to solve a problem but are getting nowhere). As others have said - email them every single time it happens and cc your manager with screen grabs - she can then see the extent of the problem.
If.. and I mean IF... it is something that a bit of extra training on your part could help with, then embrace it - approach it with a positive mindset and use it as a learning experience - then demonstrate you have taken the feedback on board and run with it.
Most people in their careers will get feedback at some point that they don't like - it's okay to be upset by it (particularly if you had no warning or indication that there was a problem - no one likes to be blindsided) - but it's then all about how you deal with it. If you acknowledge the feedback, recognise that improvements are needed and, if necessary, ask for help with it - that is usually respected by an employer. If you are negative, refuse to take on feedback and don't take responsibility for what you can do to improve things - then that's where you'll get further problems.
It DOES sound like your new manager could communicate better - there are plenty managers who have poor communication skills - but the way to deal with that is to go back to her - ask for a meeting - tell her what you need from her. Do you want more clarity on what she feels the problem is? Do you need her to help you sort out the tech issues? If you are doing everything you can from your end - and can demonstrate that - then it at least puts you in a better position professionally and gives you grounds to take it to HR if it IS simply down to poor management.
You can sort this - but it takes proactivity and a demonstration of wanting to solve the problem.