To answer your question, no, it isn't sinful to have a close, nonsexual relationship with a person of the opposite sex. I have such a person myself, a girl I knew in high school, who is married and lives several states away from me. I love her dearly, and I affectionately call her my sister, and I send her gifts in the mail. Her husband knows my good nature and does not worry about anything indecent going on.
The issue of "sin" comes into play when it involves lustful thoughts towards someone, or engaging in activities that would be considered sinful. I'm sure as a Christian, you know of what I speak, and if not, I have no problem elaborating on it.
However I do want to point out that based on your description, that this relationship is perhaps more than just a close, platonic friendship. You said that you were aiming for marriage early on, but you two were too scared of health costs. And that you decided to spend your time together in a chaste, platonic relationship. So a compromise.
If it ever gets beyond chaste, platonic relationship, and you two start developing romantic feelings for each other, there are two paths that I see that you can take. You can do things the right way, and get married, even if it affects your insurance coverage. Or you two may have to distance yourselves from each other temporarily if the feelings become too strong, and they become problematic for you in maintaining the neutrality of your relationship. Distance can either produce longing, or it can make you realize that you are not some emotionally dependent on someone else as you feel like when you are involved with them. Just some thoughts and advice.