It is certainly no sin to want sex with your partner, it is still certainly no sin to still crave sex even when you know your partner is not wanting sex at the time of your craving.
A libido is not something that can be just switched on and off like a light switch the OP is still breastfeeding and is not having much sleep so it's understandable her libido is low.
What any good caring and loving partner should do is say is something along the lines of hey I'm really missing our sex life and I'm really craving the intimacy we get from it. Is everything ok with you/with us? Is there anything I can do to help? Would you maybe want to see a doctor about it? Do you just want to talk about it with me? We can figure it out together don't worry.
You DO NOT tell your partner that your love for them is dying and that you will go get it elsewhere if you don't get it from them!!
Put it this way, think of the libido as an entity or an organ like the lungs or heart. If the lungs or the heart got sick or were not feeling so well or pumping away like previously, would your thoughts be of leaving it? Would you be threatening to rip it out and replace it with a newer racier model? Or would you first look at what you could do to make it better and make it healthy again so that perhaps it could work the way it once used to??? A considerate and loving caring partner would want do what they could do to to help make the organ healthy again and the partner with the sick organ would also want to do all that they could do for the organ to become healthy again.
Sex and intamcy is an important part of any relationship it should never be used as a weapon, held to ransom, be withdrawn nor forced upon. It should be natural and evolving and just like life it's ever changing. Sometimes it changes together and sometimes not. At times of mismatch we all still deserve love and care and most of all understanding. What both partners have to understand is that whilst it is a difficult period it must be talked about and worked through and it must be done with care and compassion from BOTH parties. A sexual relationship is a fragile thing and must be nurtured by those in the relationship, it's too often taken for granted, in that it's just expected to just work and just flow. Perhaps it would be easier if we talked about sex with our partners more often and started the conversation when thing were good and not just when we started finding this difficult. I think for most by the time it's talked about during the period of difficulty there is already pain, rejection and resentment and then that's when it gets hard to solve.
Gosh...don't I sound like a right hippy 
Food for thought though eh. 