So sorry you are having a hard time with LO. I had this scenario with dd1. I managed to help myself better with advice from here and NCT etc for dd2 now 5mths and still bf.
At 1 week its difficult to establish a pattern and for me took several months (3) even when going well.
It is possible that the baby just wants comfort or has trapped wind.
With dd2 who cried lots after night feeds I gave a dose of infacol before each feed after 5pm as it was worse at night.
If you can have baby in your room in moses basket/crib and feed lying down especially at night- you can doze (or fall asleep if anything like me!) while baby feeds. Hormones are released during breastfeeding which are similar to those released after sex and will help sleep.
Don't expect to get anything done! Just look after baby and get any help you can from relatives and friends. Let them take the baby out for a walk in the pram- if he cries it won't hurt if it means you can refresh and carry on. Better than you being exhausted.
At some point you may wish you had never had the baby because you are so tired- its normal and will pass. 12wks seems a long time but in a lifetime it is nothing. This phase will pass.
To settle the baby without feeding try cuddling, patting rhythmically like a heartbeat, winding across your knee instead of upright, rubbing tummy firmly but gently, changing, take for a walk, swaddle, rocking in infant carrier or swing. Try not to use a dummy yet if you can help it, it will probably just fall out anyway.
When I was tired in the first few days after a difficult labour and birth I gave formula at 2200 which made dd2 sleep for hours. That way I got rest, she got sleep so she wasn't overtired and falling asleep before she was full iyswim.Once she had sleep she fed better. I would try actively getting her to have 3 naps per day of 40 mins to stop overtiredness, sometimes the rooting is for comfort and not hunger. If your DP/DH feeds at this time for 3 nights and you continue to bf during the day and middle of the night you should get enough rest to go back to bf at that time as well, it won't work for everyone but did for me. As I have said I have ex bf since day 4 and dd2 is now 5mths.
Keep going you are doing a great job, he is thriving. If you have sore nipples use lansinoh/breast milk before and after every feed.
I gradually tried to increase time between feeds by distracting/talking/giggling etc but don't expect 3-4hrly feeding for a good few months and in my case until solids were introduced.