Agree, the peat pots are a disaster. They hold too much moisture and keep the seedlings waterlogged and then when they finally do dry out they suck water away from the seedling.
The seed starting mix is probably high in peat, compounding the problems.
And you said they get plenty of sun. Is that because they are outside? If not, if they are indoors, like in front of a window, they are NOT getting plenty of light. At this time of year you can just put the plants outside and bring them in at night if your nights are still cold. (But if they've been inside, you can't just throw them outside, you have to transition them gradually.)
Spinach is a cool weather crop and fast growing. I never bother starting it indoors. I plant the seed directly in the ground as soon as the soil can be worked, which for me is sometime early to mid March. If you have more of the spinach seed, I would start over just planting it in the ground. It is already a little late for it, meaning it is likely not to last a real long time, but bolt (go to seed) fairly quickly. So save some of your seed for planting late in the season for a fall crop, which will do better.
I'm not too familiar with eggplant, but I'm thinking you might just start that one directly in the ground at this point too. It's a warm weather crop, so it doesn't get planted until your soil has warmed up some. Which would be sometime between now and a few weeks from now.
If you have some pepper seedlings that look reasonably healthy, you could just transfer them to a plastic pot (or plastic drink cup with holes in the bottom for drainage). If you use potting soil instead of seed starting mix, it usually comes with Miracle Grow or something like that in it. That is all you need until putting them out.