There's a lot of great comments on here already. My 2 cents.
1. The first 1-2 years of parenting are survival mode. It's wonderful! It's joyful! And its full of sleep deprivation, illness, identity shift, marital stress, lack of freetime, etc. etc. Don't underestimate how much the sleep disruption is contributing to stress, illness, and general irritibality with work and relationships. The good news is that most (though not all) children start to sleep through the night around 12 months. This is a game changer for many.
2. As others have said, the constant cycle of illness is normal. Even when one parent stays at home! Babies/toddlers get sick All. The. Time. They touch everything in public places and live with their hands in their mouths. Illness will drastically reduce in frequency over the next couple of years. I swear. +++ on the ear tubes!
3. Note that the happiness curve for parents is rough the first few years. The daily grind is HARD when kids are fragile and require constant supervision. Parental and personal happiness often improves as kids get older (before taking a dive again during the teen years). Hold on! It's going to get easier! Gaining back free time, space for hobbies, consistent sleep, marital time, etc. can feel painfully slow and gradual the first year, but it builds and things open back up again. (*note that people with multiple kids say the difficulty clock "resets" on those tough years with each new baby - compounded by sibling joys and challenges. We only have one so I can only speak as a family of 3).
4. Your wife - if she's like the majority of women - is likely also grappling with frustration over her own postpartum stuff. New body. Potential decrease in confidence. Milk making (if applicable). Resentments can build for both of you as you non-verbally (or verbally) compare and contrast who has it easier/ who has more free time/ who is getting more sleep, etc. Even the most egalitarian/ open-communication marriages can suffer as you both take turns pity partying. This is normal, but I recommend you both giving each other more grace, more patience and more support, regardless of how much you're already giving and whether you think you are already the person giving more. Offer to take the baby alone unexpectedly for a walk so the other person can have 20 min to Do. Absolutely. Nothing.
Invest in help with food/laundry/date night, etc. Resist the urge to keep score. Remember, the first 1 - 2 years is Survival Mode. You're on Team Survive! The house looks like shit when you got home and it was your wife's turn to keep everything in order - are the baby and wife still alive? Excellent! A win for Team Survival! No one had a chance to wash clothes this week but you survived the most interrupted night of sleep of the week? High Five your Teammate and tell her she looks smoking with those bags under her eyes! If you run towards your partner (emotionally, logistically), chances are, she'll eventually pick up on it and run towards you too - maybe offering you a chance to sleep in while she makes the coffee. Think of it as a long game team sport, rather than an individual marathon that's stacked against you.
I totally get where you're coming from. Yes, explore some job options (in consultation with your wife). But know that new jobs come with stress of their own as you learn the norms, upskill, figure out new office politics etc. The best thing you can do is invest heavily in your health and marriage regardless of income or job. We somehow were both in school FT and working (me PT, my spouse FT) while we had a newborn. She's almost 10 now, but we look back at the first year as a blur. It makes us laugh how tired and strung out we were, but enduring it together made us stronger as a team.
You can do this. It will get better!