I'm coming from a very similar situation where my stress and unhappiness from my previous job caused me to be very forgetting and additionally very easy to get angry. Causing some small fights.
You need to do the same with your family members, as you did with your girlfriend.
Since you can't have the same long talk with everybody focus on the important parts and try to help them understand there's no ill intend in your behavior.
I don't know the specifics but if you could get a hold of a bunch of family members at the same time it would help
Sit down with them and tell them that you noticed that you are more forgetting in the past days/weeks/months and that you are truly sorry. This will make them realize you know about it, assuring them they are not alone with this conception, which sets a good start for the setting and conversation.
Explain that you are under a lot of stress and that after a day of working/studying your energy is mostly depleted, causing you to be more fatigue which sometimes results in forgetting things. You can state that you try to remember but it just get's lost under all the other things on your mind which are causing you stress, if that is the case.
If it is very important to you, you can also mention that your relationship is also of very high importance to you which might also be the cause for some occurrences of forgetting something, this can just be a little to bring across without making the impression your relationship is higher than family.
But most important of all; don't lie and don't make things up. Be honest and explain the situation briefly.
I would recommend to open the dialogue now after your introduction of the topic. Ask them about their view point on the situation, try to know what is hurting them if you forget things and try to make a plan on how to fight it with them. Including them not only in the situation but also asking them on their viewpoint will help to "smoothen" the situation between you. This dialogue doesn't need to be long, just make your introduction statements of the topic/situation and ask a question about their view point to make them feel included on the situation. That should be enough for most cases.
They might offer help, or maybe not. Also they might ask you less fetch requests for example to help lessen your stress of trying to not forget.
But even if they don't change their behavior or try to help you, they should now understand your situation and be less hurt when you forget something.
Another advice I would give is to try and remember a little bit more even after discussing this with them, to show a signal that you really try to not forget.
There are plenty of methods and tools to aid you with this.
You could try a to-do list app which allows for multiple users, syncing the to-do's and maybe even assign a to-do to himself/another person.
Maybe set up reminders in your phone/pc/mail to remind you of stuff at a certain time, like just when work ends for example.
If there are cases where the person doesn't seem to be less hurt or is still not understanding your situation, you might have to fall back to one-to-one talks which might be a in more detail and dialogue.
I wish you best of luck!