I haven't used a tarp in years, but I can remember a couple of things:

1. Site selection was never too difficult (in Ohio, Indiana and Kentucky, where I mostly camp.) The forest duff is pretty good at absorbing water, so most flat spots worked as long as there wasn't evidence that water moved across it: no erosion marks or even "streambeds" through the leaves and other debris. I usually tried to pick a higher spot if possible, but that didn't seem as critical as flat with no evidence of previous water.

2. I used a bivy sack under the tarp (sometimes with a groundcloth, sometimes without.) It didn't need to be expensive - even a "bug" bivy would work. Essentially, bivies have bathtub floors just where you sleep. My favorite was an REI model that had a large section of mesh at the head, with no closure (clearly not meant to use in the rain without a tarp) that weighed about a pound and cost about $100. As long as my bed was dry, a little bit of water running through the tarp wasn't a huge issue.

3. I made sure the tarp covered my groundcloth with about a foot of overlap all the way around. I mostly used a 4'x6' groundcloth (if I remember right) under an 8"x10" tarp which I mostly pitched in an A-frame style. The fact that I camped in a forest also helped: the trees provided a windbreak that virtually eliminate rain blowing into the tarp. (Sometimes I also had to pitch the tarp low, just high enough to sit under, to mitigate blowing rain.)

I did pay attention to site selection, but I didn't obsess over it. But, like I said, my experience was all in those 3 states; conditions in other types of climate and terrain might be very different.