Yes, you can be allergic to the bedding, to the washing powder you wash your sheets in, and sadly to the building materials that were used in your house.
If your pillow is stuffed with down, try a synthetic one, if synthetic try a down pillow.
I am allergic to anything made with or containing polyester and it is difficult to get anyone to take that seriously. Synthetic pillows and duvets are sold as "allergy free", which is fine if you are allergic to sheep's wool or duck feathers. Not so good if you are allergic to man-made fibres.
Change your washing powder to one of those tested as suitable for people with allergies or astma.
Have you tried sleeping with a window slightly open?
Are there any signs of black mould anywhere in the house?
If the house was built in the 1970s the blockboard etc. used may well still be given off too much formaldehyde, which fits some of the symptoms you describe.
Being tested for allergies is a lengthy process and may not end with you showing any allergy at all, if like me you are trouble by something the medical profession denies can hurt you.
Keep a diary of what you eat, wear, wash in or with for a month and see if there are any indications on what you don't do on your better days.
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