Advice needed about our rescue staffy x

We rescued a gorgeous staffy cross on New Year’s Eve 2014. He is our baby and an amazing loyal dog but I was wondering if anyone could help with a couple of issues we are having.
He has severe separation anxiety that we thought would subside over time as he learnt to trust us. We don’t know much about his past except he was approximately 15mnths old when we adopted him and he had been found wandering the street and was 2hours away from being PTS when the rescue centre took him on. We believe he was abandoned and may have been ill treated in the past, which would explain his anxiety. However it is so severe that if we leave the room he howls and cries and can never be left alone, it’s like he thinks we’re never coming back. It’s heartbreaking to think he feels this but also makes popping to the shops, even for 5mins an impossible task and either myself or my partner always have to stay at home. Does anyone have any suggestions on how we could combat this? We have no intentions of leaving him for any long periods of time but just being able to pop out together for an hour or 2 and knowing he’s ok would be a big help!
Our other issue is with other dogs. He gets on with 95% of dogs but has the odd dog he really doesn’t like, it’s usually dogs bigger than him, however he is best friends with my parents very large and bouncy golden retriever so I don’t think it’s just te size that is the issue! He is usually the most loving dog and will have a great time playing with others but he has shown aggression towards other dogs (never humans) and has recently got worse. He attempted to bite another staff last week at his dog training class, and an Alsatian at this weeks class. We couldn’t see that these 2 dogs did anything to deserve this! Anyone have any suggestions? We just want him to be happy and settled and have doggy friends he can play with instead of people always being wary of him! Please help us!