This is unlikely to be just bad behaviour. It sounds like he is really struggling and doesn't know how to communicate it.
Does the start of the behaviour change coincide with starting school? If so, that's a good indicator that school is where he's struggling.
Do you have ASD/Asperger's/ADHD/sensory processing issues or similar in the family? Some of the behaviour your describing is similar to that parents see in children with ASD.
I know school resources are stretched but the staff need to find time to observe him and identify triggers to the behaviour in school.
You need to do the same at home so start making a note of things like the times of day things happen, whether it's around a change of routine or an unexpected change of plan, whether he is worse after busy social activities or experiences of high sensory input.
All behaviour is communication. You should probably use the very minimum in terms of sanctions (especially) and rewards. Focus on removing the distress that causes the behaviour, rather than trying to find ways to make him hold his emotions in.
Teach him about his emotions by labelling them for him; "I can see that you're angry", "I know you're disappointed and upset about this...|"
Give him strategies for releasing his anger like punching pillows or running around the garden.
Try to allow him to move and make sure that he is supported with movement breaks, fiddle toys, a gym band on his chair or a wobble cushion at school. Children with a poor sense of proprioception can need to move in order to concentrate.
Give instructions one at a time. I know this takes longer but it sets him up for success and will reduce the times adults are cross with him. Google executive function difficulties to understand why he finds it hard to follow instructions.
If this doesn't resolve quickly, or school deny that he needs any support, go to your GP and ask for him to be referred for a neurodevelopmental assessment. The waiting list will probably be long and, if your concerns turn out to be unfounded, you can always remove him from the list. The assessment would help with strategies to support him at home and in school to reduce his stress, which will, in turn, help his behaviour.