6A81 is a carriage path error related to the print head which is the part the ink cartridges sit inside of. the next step would be to try a print head cleaning kit like this one: Click Here
It says Brother but it works with all brands as all kits come with the same thing. It comes with solution that dissolves dried ink that may be on the sensors or in the ink line a syringe and instructions. If this still does not work it is it means the print head has failed. In order to fix this the print head inside the printer needs to be replaced.
I really hate to have to tell you this but after the troubleshooting we have done if the problem is still there I am sorry to say you indeed are going to require a new part. I know this is not what you want to here but I want to make sure I am honest with the information I provide you. You would need to replace the print head in order to fix this issue.
I wish there was another way but with everything you have posted I am confident this is the problem with your machine. The reason for this is he print head is the part of the printer that puts ink from the cartridges onto the page. Over time these parts deteriorate and need to be replaced.
A new print head normally costs between $60 and $80. It would take about 30 minutes to an hour for a tech to replace it at their shop rate. Most Shops charge between $65 and $75 per hour. You can get the part from here: Click Here
In case you would like to look for it somewhere else the part number is XXXXX
Due to the price some people do decide to replace the machine if it is more than a year old if you decide to go that route I would be happy to help you find a replacement.
If this machine is under a year old Brother will normally cover the part under warranty. If you have any questions or need any help following these steps click on reply to expert and let me know. If not please consider leaving a positive rating for my help it is greatly appreciated.