I think Dawn has some very interesting points - as usual, her answers are well thought out.
I think you and your husband come from such vastly different backgrounds, you haven't really worked out a compromise. He's saving for eventualities, and everything is on a "wait" basis. Wait until we really need it, wait until there's an emergency. You want things NOW because you always had luxuries. However, I see that you are not extravagant and you are not frittering away your money on Starbucks etc. You want a dryer. And yes, chocolate is a necessary health expense!
It would seem that you could meet halfway. There is probably plenty of evidence that an energy-efficient, front loading washer would save electricity and water, which should appeal to your husband's frugality.
I think counseling would help you get on the same page. I think separate accounts can be helpful - each of you has your own account for your own expenses, and then there is a big household account for the joint expenses (mortgage, utilities, food, insurance, kid's clothing, etc.). You should be able to spend money without asking him, although it would be nice if major appliances were agreed upon.
It concerns me that you are considering getting another job - on some level, family time with your child should trump possessions. It also concerns me that you use the term "allowance" for your phone bill. It's kind of like the 1950s when Lucy Ricardo got a household allowance from Ricky, you know?
I think I would pay the nanny for an extra hour and get some couples therapy with your husband, to strengthen your communication, express your individual fears and values, and work out a solution. This situation is going to get more difficult as your child's expenses rise (school or other activities that cost extra, college applications and fees, braces, you name it). I think you and your husband share a lot of love, but not the same values, and you have different fears. Those need to be addressed in a spirit of calm, cooperative sharing.