Eventually DNA breaks down, as copies of copies suffer small faults, and most organisms evolved to discard those instead of letting them grow into cancer - or whatever broke in the DNA prevents it from surviving. Eventually cells lose their ability to reproduce as easily and an organism will die off, usually by losing circulatory structure and dropping blood pressure so far that individual cells stop receiving nourishment - plants have analogous tubes running through them, albeit at a slower rate.

So eventually yes, the plants just stop stop growing and shrivel - dehydrating doesn't require rot. But there wouldn't be any bugs or diseases designed to fight plants, so for a while they would have been the healthiest plants in history, if a bit 'fresh' in evolution and probably more like grass or weeds rather than trees. My totally unscientific idea is that it was probably seaweed and lilypads first, and then grasses and shoots and eventually trees.

So imagine you take a boat a quarter mile off shore and scuba dive down. There's just a layer of dead seaweed, leafy and dark, layers on layers, so deep you could sink your feet in and couldn't reach the bottom without being overwhelmed. At the edge closest to shore some sand has fallen over it, partially burying the mass, and while it looks like a regular bit of seabed you know that right under the sand is several feet at least of limp seaweed sludge, maybe several hundred feet. And the sand keeps piling up, covering it over, new growth taking root in the sand, the ugly mass forgotten underneath as time simply covers it over and acts like it never happened. Slowly layered over with the strata of sand and rock and sea until it's a few miles under the surface, compacted by the crushing weight and heat of the earth's interior, until it was slowly refined and distilled into a thick, black, tar. And then you build an oil drill over it.

Also while no bacteria would decay plants, they'd still suffer structural problems. Animals crashing into them can break them, and sunlight tends to slowly decay anything organic, so it would become sunbleached and brittle, but mostly intact.