The weather this year has given us "slow starters" a window of opportunity. Everything you mentioned, with the exception of the tomatoes, can and should be direct seeded in the garden. I fear you will have to buy tomato plants, unless you know someone with too many seedlings. Perhaps you could work a trade, seeds for tomato starts.
If it were my plot, I'd lift the landscaping cloth, spread the compost, plant, and then use the leaflitter and mulch to keep weeds down as much as possible. It is going to be very difficult to judge how big a hole to cut in the cloth for a mature plant.
Your tomato seeds can be started in late June for a fall garden. Your county's agricultural extension agent has a list of vegetable varieties suitable for your area. This free list will have the best Texas-friendly tomatoes on it.

my mom has been gardening for 20 years and told me the exact same thing as marlingardener said regarding seeds about my seeds last night when I was talking to her on the phone. I was going to still try to start from seed anyway but I think I will avoid the future frustration and disappointment by buying young plants for my spring crop and starting from seed for a fall crop.

Yes, at this point, you would definitely want to go with plants. If you were to plant seeds, you are looking at 6-8 weeks before you could even get them in the ground. Which would put you late April/early May. They wouldn't have time to produce before the heat sets in.
There are some really good nurseries around DFW that have an awesome selection of mater plants, both heirloom and hybrid.

Susan

It may be too late to start tomatoes, but I always start my cucumbers, squash, beans, etc. inground after the soil warms a bit and the days stay warm enough to germinate, and I get an abundant supply of veggies each year. I do water, but you would have to do that anyway if the rain stops.