Congratulations on No3!
A happy mummy is necessary to have happy children. But perhaps there are other avenues to explore - it would be very sad if you gave up, thinking it would sort out your problems, but then found yourself just as bogged down with bottles etc and no better off than before. IME the only hassle with breast is that it has to be you doing it, and that it always takes at least one arm. OTOH bottles are even more hassle as they take both your arms and there's usually more winding and yuckier nappies, plus there's washing, making up, sterilising etc.
I may get slated here, but what about trying to lengthen the gap between feeds once or twice a day - by using a dummy, for example? Or perhaps you can carry her in a sling so that you can feed one-handed? Also, there's no harm in the occasional bottle, and your older children may be entranced by the opportunity to feed their baby sister themselves.
Can anyone help by taking your older children out? Of course you want to dedicate time to them, but a playdate will please them more than a frazzled mummy.
Ideally we don't want our children glued to the TV or PC, but this isn't an ordinary time, and an hour or so a day can be offset by a visit to the park - hopefully you'll be able to feed the baby during either of these activities when the older ones may not be as demandig of your attention.
I think that maybe if you don't feel as pressured over the feeding, you may not feel as robbed of your body as you do ATM.