This stage is hard for parents, and especially for those planning on homeschooling. We think, yeesh! if I can't even get the dishes done, how am I gonna homeschool? That adds to the pressure.
While preschool is no big deal in some ways, I will go out on a limb and be the (so far) sole advocate for skipping it. Learning to occupy yourself is a great skill. Your dd2 isn't old enough to play much with her sister, but my kids are that age spread, and they are best friends now. The change happened around when dd2 was 2 and could play along in her own way with dd1's creative games. Before that, dd1 only saw her sister as a bother.
Of course, you could look at my house at decide to reject my advice entirely! But 6 and 4 (my girls' ages) is a universe away from 1 and 3. Completely different dynamic. To get stuff done, I often do some exchanges, I read some books and let them know that if they want my company, then after that they need to help fold laundry, or at least play in the bedroom where I do the work. That is one example. I've been trying to give them more "grown up" things to do, like flipping the pancakes with supervision instead of just mixing the batter. Putting in the soap and pushing the buttons of the washer instead of just folding laundry. If 3yo can play a computer game, then they can operate appliances, they can use a digital camera, etc. In the end, somedays nothing much gets done. This will pass. Or at least evolve!
One last point. We are unschoolers, and preschools don't operate like this. Preschool today is much like kindergarten "in my day" (does that sound old or what?) If your tendency is towards unschooling, or at least delayed academics, then preschool might not be the best choice. If you think you'll lean closer to playing the role of teacher at home, this experience will likely fit right in. But I would be extremely careful about the preschool. Some are really pushing desk time over playtime. Academics might be diverting at this age, but it should not take precedence over playtime. And they need this every day! 3yo is when the imagination in playtime really blossoms. The "baby games" of putting-in-taking-out-tucking-in will move aside or change into long running stories you will delight in hearing. Free play rules the day. It is necessary. If the preschool doesn't offer this REALLY FREE play I would personally skip it.
So, I would give it time. What you are experiencing now is a struggle. But in not too long your girls will be playing and be best friends (and *ALL* that comes with that, let me tell you!) This is an asset, as they will be spending a lot of time together when you homeschool. Having Big Sister around has really helped dd2 become the girl she is today.