Is it permissible for me to address my husband's father as "my father" (أبي) and his mother as "my dear mother" (أمَيْمَتي) with the intention of showing affection and tenderness of hearts, or does this contradict the Almighty's saying: "Call them by [the names of] their fathers; it is more just in the sight of Allah"?

There is no impediment for a Muslim to say "Ya Abtah" (O my father) to an elder who is not a relative, if it is done as a form of respect and endearment. This falls under the command to "speak to people good words," and the prohibition only applies to a person attributing themselves to someone other than their true father, as a way of disavowing their real lineage.