The carers allowance that you get doesn't make a difference in this situation; it is based on mobility allowance.
The road tax exemption mentions that people on Standard
Mobility PIP (50%) are eligible, this is as low as it goes so if you are on lower than this you would not be able to claim.
However, to be sure I would recommend calling the Disability helpline on 08457 123 456
It does say that the exemption certificate is sent to you directly if you are eligible so I think that might rule it out in your case, but it is always worth checking.
As for the blue badge, you should be eligible for this but you need to apply and I have to tell you, the form is very long and involved. It is definitely worthwhile though.
Now, when you say you get carers benefit, do you mean that you get benefit for caring for someone, or does someone else get benefit for caring for you? The reason I ask is that although they may be able to apply for the badge on your behalf, it is strictly only to be used when you are in the car.
You can apply for a blue badge online here https://www.gov.uk/apply-blue-badge
If you have any questions, or would like a form sent directly to you, contact details are as follows:
Blue Badge Initial Enquiry Support Service
Telephone (England): 0844 463 0213
Telephone (Scotland): 0844 463 0214
Telephone (Wales): 0844 463 0215