that cowltop vent feeds the louvered kick panel vents at the sides, it's unlikely any water is getting under the dash directly from that, but it comes off easily with a few screws [under hood edge] once you remove the wiper bezels and you'd then be able to see for yourself ... it's entirely possible the cowl side plenums are rotten, as most get to be, and that could allow seepage from the sides onto the floor where the mat would spread it across the floor, especially if you have carpet or the old style underlay below a rubber mat - take off those inside vent covers to judge that.... it's also possible the windshield rubber is letting water in, especially if it's old, and the flange it mounts to could corrode ... another possibility is under the trim piece over the defrost vents - at the ends there's holes that drain any moisture into the cowl plenums, and they rot as well, could pop that strip off [screw at each end then about 6 spring clips across] and see what that looks like
the door seals will likely never do very well at keep all water out because of the poor dogleg design at the front never fitting real well, out of half dozen TF trucks I haven't had one that didn't leak down on the step from the dogleg
sorry for the bad news, but the TF series is another example of cheap consumer goods meant to do one job [hauling loads] not being very good at anything else

also, covering the truck with anything resembling plastic before putting another cover on it is a sure bet to ruin your paint job, if that matters
Bill