Hiya!
I had this exact same problem, and I mean my hair was carrot orange! Sadly if other techniques haven't worked, your best bet is to buy a dark brown hair dye and apply that. It WILL NOT apply evenly if you have very bright orange roots, but it will just neutralise it to different shades of brown (my lighter orange bits actually turned a nice light brown and people thought I had gotten highlights done!)
In order to not dry out your hair I recommend you get a dye WITHOUT bleach. Most dark brown dyes don't have too much of a lightening agent but actively avoid ones that come with a bleaching product AND a colouring product. If you are willing to risk it like I did - the best bet is bleaching your hair. I can understand why you would not want to do that however as its very risky at home. I have very short hair so bleaching it has always been easy - I understand its much more daunting with longer hair.
Here is a quote from an article I used on how to get the best hair dye:
"Color selection is vital. Avoid colours that contain a red base, which are typically described with words such as warm, chestnut and mahogany. Ash colours contain a green base that neutralises red. Words to look for include cool and ash. Always use cool, ash colors to cover red hair colour if you don't want to see any red tones peeking through.
Hair colour starts to oxidise as soon as it's mixed. As hair colour is oxidising, it continuously gets darker. A medium-brown colour may oxidise to the point that it appears black, but it's no cause for concern. Your hair won't turn black as a result of oxidation."
All the best! Hopefully you won't have to dye your hair but if nothing is working here's my two cents xx