Of the 7 examples of 'like old books' in the Google Books Corpus for the 1920s to 1940s (there are none after that), only two are similar to yours:
.. I had to promise solemnly that there should be no marriage for three years, and that I would stick to my work like old boots, and make a position for myself first."
... know what it's been like this last year since I had that row with Dad about poor old Willum. Tom's going to be allowed to go into the Navy where he wants to go, but I'm the eldest and I shall have to stick on the farm like old boots ...
It appears to mean 'very firmly'
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