If you're strictly refusing the inclusion of "romantic relationship", then the closest term to what you are looking for is as follows:
"A romantic friendship, passionate friendship, or affectionate friendship is a very close but typically non-sexual relationship between friends, often involving a degree of physical closeness beyond that which is common in the contemporary Western societies. It may include for example holding hands, cuddling, hugging, kissing, giving massages, and sharing a bed, or co-sleeping, without sexual intercourse or other physical sexual expression." (Source)
Basically, it's those friendships that you could mistake for being lovers. You know the ones I'm talking about. It's the friendships where they act like a couple but they aren't.
Alternatively, there is also the more obscure and significantly more confusing...
A QPR "is one which is more intense and intimate than what most people regard as a friendship, not fitting the traditional romantic couple model or the traditional bounds of friendship. It can be characterized by a strong bond, affect, and emotional commitment not regarded by those involved as something beyond a friendship. It is a so-called platonic relationship, so it does not comprehend sexuality/eroticism or romance, although some people involved in light or non-traditional romantic relationship might also categorize themselves as being queerplatonic." (Source)
Personally, I hate the way it's described in that, but that can work for what you're looking for. That said, despite what this Wikia says, QPRs CAN still include ACTUAL romance, (ie wining-and-dining), it just removes the pretense of sex from the relationship altogether.
That said, QPR's an awkward term that really fits better for people who are in "closer-than-friend" but "not-quite-romantic" relationships and usually for people who are asexual/aromantic.
Honestly though, saying you're in a plain-old "romantic relationship", "nonsexual romantic relationship", or "nonsexual relationship" should work just fine and will probably make more sense to explain to people than either of these options.