PLEASE HELP ME!
You didn’t say where you live; that would help. It isn’t helpful to water every day as the soil needs to dry out a little to breath. Did you punch holes in the bottom of the cups for drainage? If not, you are killing your plants with too much moisture.
They probably need more light than you are giving them. Those plants are Full Sun plants, not like a houseplant which will thrive on a few hours of sun a day.
Look into getting a book on plants or contact your local University cooperative extension Master Gardner’s program. They will offer FREE local advice and assistance to help you and identify your specific problem.
I think you should get large self watering pots (Wal Mart has them really cheap) and garden potting soil and some plant food and transplant the strongest looking seedlings into these large pots and leave them outside if you are past the frost where you live. If not, then you will need to take them in at night.
Don’t put them in dixie cups. The chemicals in the paper will get released when you water them, and plants don’t like chemicals, usually. And dixie cups don’t have much room for them to grow roots. To save them, transplant them outside, now. You’ll want to get a trellis for the cucumbers to climb on and the tomatoes to lean on, and some fishing line to tie the tomatoes in place. Put them in the sun and keep watering them daily, and they should be fine.
treat them like your kids , water them softly and easy always place a plant the same direction to the sun they go through this crazy thing to reposition to the sun.they do get confused.
I think your Dixie cups are fine as long as you punch holes in the bottom for drainage, It seems yours are drowning. The starter plants need 12 hours of sunlight per day. If that’s not possible then they should be placed under fluorescent lights for 16 hours per day until it’s warm enough outside to transplant to permanent pots. If that’s awhile yet you’ll need to transplant from the dixie cups to larger containers — at least three- to four-inch pots so the roots have somewhere to expand.
Good luck!