Dear Nickers,
You are doing an excellent job, but it is no wonder that you are exhausted after the last two weeks, and with another little one as well. Only you will know when your limit has been reached. but, I can tell you that, with the right care and support nearly every baby will breastfeed. it's what they are designed to do. But some need some time and support. There are usually a combination of factors that cause a baby not to feed - and believe me, it's not that he doesn't like your breasts, it's that something is making breastfeeding difficult for him.
As hoobley said, sucking is a developmental stage. Just like some babies walk at 9 mnths and some don't walk till they are 16mnths, some babies will breastfeed vigorously at 36 weeks and others won't "get it" for some time later. The other important thing is, that at this age you don't just want to write off things because you have already tried them. Every day will bring change and development. The nipple shield that didn't work last week will work beautifully this week (or next!) You baby may need some time to get over the birth. Following his lead will often be important. Try to observe his cues for feeding to catch him just when he is most "ready' to feed.
Have you got anyone who can help you out with your older child? It's worth calling in your favours with your partner, friends and relatives to help you get breastfeeding right.
Try this. Get yourselves comfortable on the couch, or in bed. Undress your baby down to his nappy. Take off your top. Just let your baby lie on your chest, skin to skin . There need be no pressure to feed. You may repeat this several times a day, and don't be disappointed if nothing happens.
Just let him rest, maybe give him a little back massage. When he wakes up, you may notice that he bobs his head up and down on your chest. He will bring his hands up to his mouth and suck. Just let him do his thing - it's all part of him getting himself organised to feed. You may notice that he eventually kind of throw himself sideways at the breast. Gently assist him to get the nipple. Once again, you might find it handy to have a nipple shield handy. Even if it didn't work last time, it may work next time. Avoid pushing his head onto the breast. Babies hate that!
If you send me a private message with your location, I would happy to recommend an LC in your area. Sometimes all these things work better with gentle guidance - it's a tough gig by yourself with no real roadmap
Goo luck. let us know how you are going
Regards
Barb