NEW YORK — Last month the publicity-shy David Letterman paid a visit to the CBS affiliate in Dallas, KDFW-TV. During a two-hour stopover, he met the station's executives and staff, tossed a football with some passersby outside, did shtick on the air with the KDFW weatherman and videotaped some local promotional spots for the station to air when Letterman's new late-night talk show debuts in August on CBS.

It is a measure of the stakes--personal, financial and corporate--in the push to get CBS affiliates signed on with the new Letterman show that network executives asked him, and that he agreed, to show the flag at KDFW and two other stations, in Houston and Raleigh, N.C.

Luring Letterman away from NBC with a reported $14-million-a-year contract was only the first hurdle CBS had to clear in its effort to establish a late-night franchise that will challenge NBC's long-running "Tonight Show" and ABC's "Nightline." Now the network has to persuade its stations to carry the show--and at the preferred hour of 11:35 p.m. (an hour earlier in the Midwest).

Currently, fewer than 70% of CBS' approximately 200 affiliates choose to broadcast the "Crimetime After Primetime" series that Letterman will replace, and only about half of those at 11:35 p.m. The majority instead fill that lucrative time slot with Arsenio Hall's talk show or some other syndicated series. KDFW, for example, carries reruns of "The Golden Girls."

Persuading them to take Letterman at 11:35 is of crucial importance to the network because the audience decreases as the night grows later, and the biggest ratings--and thus the largest advertising rates--are generated in the 11:35-12:35 period. Only by getting high clearances there will CBS be able to quickly recoup its investment in Letterman and compete with "The Tonight Show," which is carried on 99% of the NBC affiliates, nearly all of them at 11:35.

Since signing Letterman in January, therefore, CBS' affiliate-relations representatives have been crisscrossing the country, making their case for the Letterman show. They are citing his strong ratings with young viewers and projecting advertising revenue versus other programming; they are promoting loyalty to the No. 1 prime-time network and stressing the long-term payoff to stations if Letterman is a success.