To get back on the board, you can sort of lie on the board as you are righting it and as it flips over (to be the right side up), it takes you with it. ie instead of righting the board and getting on, you sort of hop on as you are righting it using your legs. It takes a bit of practice. see here at around 1.14
https://www.youtube.com/watch?v=RrNqhL8Kynw It also helps you get back on the board much faster
Another thing I have found to save your arms is to see if you can do something that you currently do with your arms in some other way. For example, if you are paddling and then want to sit up, instead of pushing with your arms, dangle your legs and let your legs provide leverage. Only a minor difference but it can add up over a longer session.
Turtle rolling itself shouldnt tire out your arms too much - are you death gripping the board with rigid arms? See if you can relax your arms a bit (grip needs to be strong). That video linked above I dont quite agree with in that I find if your hands are closer to the nose, then you can bring the nose under much more easily and, once the nose is down, the rest of the board will generally be fine. So check that you have your hands forward (personal preference of course, this may not work for you)
In the white water, try really turning ie from going left to cutting back to going right (180 turns) or from going forward to facing right back out to sea. Practice different weighting - right on the tail, see how far forward you can go etc. Try turning from those different spots. Try different stances - make sure your hands go to the 'right' spot (eg front hand to the rear, not both hands out in front), bend your knees and crouch very low, stand up straight etc. Try moving rail to rail as you are going straight (it wont do anything but you can get a feel of making side to side weight adjustments). These may not necessarily 'improve' your skills but they will give you a better feel or understanding of the board, which will help a lot when you are 'under fire' on the wave.
If you are nose diving on the green waves, probably you just arent paddling hard/fast enough for the faster bigger waves (or, technically, its not that the swell is necessarily moving faster but the bigger the wave the more water is coming up the face in a shorter period, and you need to paddle to overcome that water movement). Positioning for the wave may also be a factor - eg because you need to be moving faster when the wave comes, often you actually have to go back slightly further and then start paddling earlier so your speed is up when the wave arrives (because bigger boards take a while to get up to speed). You may also be feeling a bit hesitant because of the bigger wave - got no magical advice for that, other than to say 's/he who hesitates, nose dives'.
Hope that helps!