Hi @SAHMofone. I completely relate to this. I am also a stay at home mom of one and my daughter is 6 months old. I was just about to post something along the lines of "feeling discouraged/needing encouragement" when I saw your post.
This morning I had to reschedule an appointment because I couldn't get the straps on the car seat adjusted. It might not sound like a big deal, but I ended up crying about it. It took me 2 hours to get out the door for the appointment, it will take 2 more hours when I have to do it all over again in a couple days, and I just really thought by 6 months postpartum I would at least have enough of parenting mastered that I could do simple things like use a car seat. My husband is at work and so I was going to text my best friend about it to vent, but I already know that she won't be empathetic about it and if anything will be a bit annoyed. When my daughter had extremely severe colic a few moths ago and I would tell my best friend that things were really hard, she would say things along the lines of how I just needed to get out more. She told me that if my baby's needs were too much for me, then I just needed to spend less time trying to comfort her, because plenty of people have more than one kid and their kids do fine without getting their undivided attention. Those kids of reactions make me feel like I have no one to confide in - besides my husband - when all I'm looking for is a little empathy from another mom.
So no, I don't think you're being overly sensitive, and thanks for sharing what you're going through. I know what YankeeCat said is true about how being a mother is hard whether you're a working mom or a stay at home mom, but I don't think you were trying to imply that being a stay at home mom is any harder by mentioning that you stay home. It's just different, and working or staying home both come with their own drawbacks and benefits. The isolation that comes with being a stay at home mom can be especially poignant, as well as the struggle to find time to pursue your own goals or have a moment to yourself or among other adults. (I also relate because, like you, I don't have a babysitter or any relatives around who can help out, and my husband is great but he's often working or out doing the shopping, laundry, etc). So you would really hope that in the short bits of time you do end up socializing with another adult, that you would feel heard, and it can just be very discouraging when that's not the case. Even though it doesn't feel like it, I think there are tons of women out there silently dealing with this kind of thing (silently because if they're socially isolated then by definition they feel like they're alone!). But we're not alone.
I guess our friends and family members often just end up in different circumstances than us and as a result they have trouble relating. Both my closest friends are working moms with more than one kid and their kids are naturally good sleepers who never had colic, etc. It makes me feel like they're supermoms. When their kids were younger they did things like breastfeed while hiking, and I'm sitting here just trying to breastfeed while staying awake. It's tough when the people you thought would understand you the most - because you love them dearly and on the surface their lives are similar - actually have very little empathy. I think sometimes they feel they know what we're going through because they feel they're been through it and they're also looking at certain similarities you share. But they haven't lived your life and I'm sure if they were in your shoes they'd be having the same struggles. I think the feelings of stress or inadequacy are natural under those circumstances, as well as the sorrow over not feeling validated by your sister, but that in reality you're doing a fantastic job and one kid or not, you're being a supermom juggling so many responsibilities that ultimately fall on you alone.