Greetings, welcome to the Forums.
By evergreen 'Amaryllis' I think you may be referring to the genus Hippeastrum rather than the genus Amaryllis. Though some species and cultivars of Hippeastrum do seem to require a Winter dormancy, many of the modern cultivars will often remain evergreen, at least if kept in mild conditions.
The two species in the genus Amaryllis, native to the Western Cape, also exhibit seasonal dormancy. In this case, it is Summer dormancy.
I have several cultivars (mostly red) planted in the ground in my USDA Zone 9B/10A garden and they often remain somewhat evergreen in winter. This year we have not had any frost aso far, and I can see healthy Hippeastrum leaves in my flower beds.
I'm not certain if there is an an 'old-fashioned' pink cultivar that is/was particularly renowned for staying evergreen, but perhaps select a modern, pink-flowered Hippeastrum cultivar that you like and try growing it as a houseplant year-round. It might stay evergreen for you.
Note, there is a green-flowered cultivar called Hippeastrum x ‘Evergreen’ (Spider Group), which has received an Award of Garden Merit (AGM) from the Royal Horticultural Society (RHS), but the name 'Evergreen' refers to flower color, not to leaves lasting year-round, though they might indeed do so...