Does reading an old letter from my husband, dating back to a previous relationship, count as spying, and am I sinful for doing so, and what can I do to get rid of the anxiety and doubts I have about him?

Whether your action is considered espionage or not depends on how you obtained the messages. If you deliberately sought them out, then that is blameworthy espionage. However, if you stumbled upon them unintentionally, then there is no blame upon you. The crucial factor is your husband's current state: if it is not clear to you that he is continuing an illicit relationship, then the default assumption is his innocence. Do not trouble yourself with obsessive doubts. If his continuation of the relationship is proven, then advise him and remind him of Allah. If he does not respond and you are being harmed, then you have the right to seek a divorce.