There is probably something jammed or broken inside
Can you take the tray out and look inside with a flashlight at where it would be picking up the paper?
Also if you pulled the tray out to remove the jam it might not be inserted quite right and that would affect the feed as well
Where in the machine is the front edge of the paper that gets jammed?
What is probably happening is that the third piece of paper is not being picked up from the tray at that point the printer shuts down leaving the other 2 where they are at the time
It might be the tray itself that's not right. Is there some sort of spring or mechanism to lift the paper so it feeds properly. That might be broken
It is hard for me to diagnose well from here.
but either its not feeding from the tray correctly or there is something jammed under the print head. Its not enough to stop every sheet but it stops some of them.
i haven't got one here to look at. But if it looks like it should be lifting the paper than it probably should be lifting the paper somehow. either a spring or some usually more expensive printers actually jack it up with a wheel from the printer side.
You can buy a new tray from brother for 20 bucks
The rollers that pick the paper up from the tray can also go bad and the paper slips off. Those might work better on the first sheet than the next one. You can purchase and replace them pretty easily they just snap on. But look at the ones on it now first and see if they look worn out. they should be a little bit tacky so they can get a grip on the paper
I will opt out. I think that you need a new tray. I gave you the link to buy one for 20 bucks.
If anybody else here thinks differently they will post in.
If it turns out that i was correct i hope that you will give me a good rating.
Your rating is what gets each of us credit with the site.