Personally I don't think it's a bad thing to get upset in front of them, it shows how much you care and that can only make them feel loved. Not going over the top but to show that you will be upset and not just be practical about it all is not a bad thing. I do wish I'd have shown my dad a bit more. I was practical, thinking I was helping and dealing with what we had to deal with but a few tears to show him how much he was going to be missed would have reassured him and I would now know that he knew how much I loved him. Anyone can say it, it's showing it that counts. I did hug him, of course, but a display of emotion would have been nice. I tried to keep him laughing.
It's very hard because you haven't done any of this before so it's alien to you. Try and get all the help you can with this. Have you any family members or close friends you can confide in and share your worries (apart from us on here)? Try Age UK, they might be able to help. Or look up any other organisation which you think might be appropriate. But it does seem like it is a box ticking exercise that they send to all people your parents ages. The trouble is, they don't realise that people are humans with feelings and not robots. Take care and look after yourself so that you can look after them. 