No, you are not unreasonable to take your children to the doctors if you think they are ill.
My story (in brief). Daughter at 9 months had a high temperature and what we thought was an ear infection. She was given amoxycillin. This is fine and normal and in most cases would solve the problem.
At 10 months, she'd been struggling with a fever. I took her to the doctors on the Thursday and they checked her thoroghly and send me home finding nothing. On Friday I wasn't happy with how she was so repeated the experience. On the Saturday morning, she was in my bed at 5 blazing hot. I turned the light on and noted she was covered head to foot with a rash. Now she springs a rash with every cold she gets so I'm very experienced in the blanching test. They blanched fine, but she wasn't 'right'. I called NHS direct and they told me to take her to OOH which is at the local hospital.
We got there an hour later and she was now quite distressed. I took her into the consulting room and showed the doctor the rash. He told me if she was his child he'd be inclined to wait an hour. He also told me that she wouldn't be this responsive if she was really ill - but to my mind she was so much lower than she usually is; very clingy, very whiny - not even vaguely herself. Then she went floppy. Then I noticed that some of the rash wasn't blanching any more. He decided to call the children's hospital for a consult and we sat there for a few minutes watching this sodding non blanching rash erupting all over her body. He made another call, this time saying it was an emergency and he needed paediatric help immediately. By the time we got her to the treatment room in the children's hospital (about 3 minutes later), her hands and legs were very dark purple. The doctor wouldn't leave nor let the nurse leave 'just in case'. She was pumped full of saline and she seemed OK. About 6 hours later I noticed that her legs were purple again and we had to go through the whole bloody lot again. She was given a cocktail of fairly hefty antibiotics every twelve hours.
She was amazingly discharged the next day as she seemed to be getting better, though she still had a cannula in and needed to go back for more antibiotics.
Now that's just half of the story. Here's where it gets relevant to the OP - sorry, I know this is hideously long.
The next 4 months were hell. She had fevers more often than she didn't. I think the longest I had between doctors trips was about 10 days. They could never find anything wrong. It got to the point where the doctors were saying 'it's teething' and I would say 'she's been on Ibuprofen and paracetamol for a week - that's not teething' and they'd say 'there's nothing wrong.' I sat there and made one doctor go through her history and pointed out this much fever wasn't normal. She decided I was a 'worrier'.
Nursery wouldn't take her because she was in such distress that she needed one on one care. I also had to work full time, and the doctors couldn't find anything.
Eventually she got up one morning and her left ear appeared to have slid down the side of her head. I recognised that wasn't right and took her to the doctor, asking Husband to check with the nursery if it had been knocked (didn't think so, they're excellent at reporting these things). Doctor assumed it was a skin infection and gave me antibiotics and instructions to watch like a hawk for signs of septicaemia. Joy.
Two days later it was no better and finally, joy of joy I was referred to the children's hospital. The ENT expert diagnosed 'mastoiditis' immediately (this is very rare in this day and age). She was operated on that evening to remove the massive cyst in the mastoid bone (the removed the bone).
And thus ended the weekly doctors trips. There also appears to be notes on her file that she is seen immediately due to her now extremely poor immune system and notes to say 'listen to the mother'.
Yes, I've been back in a panic when there has been nothing wrong but I've never since been made to feel small, or like a hypochondriac freak when I know there is something wrong.
Now the disclaimer; this does not mean that there is automatically something wrong with your children (the symptoms sound like a nasty virus but I'm not a doctor, nor have seen them). But I'm very, very aware that if I'd have accepted the 'teething' diagnosis Claudia would not have lived.
If you are not sure whether a doctor is necessary, you could ask to have a phone call, or speak to NHS direct. The worry is hideous, but it is not wrong, and it is absolutely not unreasonable. Ultimately, all of us are our children's first triage. It's up to us to work out whether they need medical help and to get them there if they do. Some of us may be more overcautious than others, but I would never, ever, ever say that a mother shouldn't go to the doctors if she fears for her child's health.