Your situation sounds very similar to mine. My kids are a little bit older, but they've all been in school full-time for 2 years. I work 3 mornings/week during the school year (will be about 13 hours/week this upcoming year).
I will be honest- I LOVE the quiet. I relish it, I delight in it. But that might just be my personality. I'm a social person- I love throwing parties and getting together with friends, but I also adore having time to myself to recharge.
As for the lonliness, I don't feel it often. I have two full days off each week. At least one of those days is my errand day- I drop the kids off at school, spend an hour or two at a coffee shop or Barnes & Noble until the bigger stores open. In the morning I do fluff/misc shopping, go to the library, whatever I feel like doing. Then I meet DH for lunch. After lunch I do my grocery shopping and then head back to pick up the kids from school. My other full day to myself sometimes I spend at home catching up on household chores, other times I'll meet friends or DH for lunch or coffee. On the 3 days that I work, I generally get home between noon and 1pm. By the time I have lunch, I've only got an hour or two before I need to pick up the kids. I spend my afternoons cleaning up the house, doing laundry, and prepping dinner. If I'm caught up on those tasks I read, play music, surf the internet, watch something on Netflix, or talk/text with friends. I also try to make sure I'm moving my body on a regular basis. I've done all sorts of things, from using the treadmill & lifting weights at home to going to formal exercise classes. My kids are involved in a fair number of after-school activities, so I sometimes don't have time to do those sorts of things in the afternoons or evenings.
If I am feeling lonely, I make an effort to spend more time with other people. I love having lunch dates with my DH. I think it's fun to meet up with friends for coffee, even if they are SAHMs and bring their kids with them. Sometimes I bring my lunch to work with me and eat lunch with my coworkers who work full-time. And I also get together with my full-time working friends in the evenings or on the weekends. I have a circle of friends who get together every Friday night for dinner and drinks (moms, dads & kids included).