As noted you need the USSR's space program to be more than it ever was OTL to start with. Keep in mind it was essentially run by the military which saw no real advantage to having a Lunar program. You also have to avoid 'embarrassing' the US into spending whatever it takes to 'beat' the Russians because in the end the US can always afford to do just that whereas the USSR can not.
The main issue is to try and keep the 'race' as even as possible within the means of the various players so that the Soviets landing on the Moon is "just" another step with the US close behind. In most cases the 'race' being closer and being less in need of a 'big' step which OTL was the Lunar landings. In such a case you'd see initially more LEO efforts with the first 'space station' being the goal followed by slowly extending effort towards the Moon.
NASA wasn't inevitable but it was highly likely as even without having Eisenhower's (justified) worry about the military opening of another 'front' in the Cold War, (note how the USSR didn't reciprocate the gesture and retained military control of their space program) there were many other factors that drove the creation of a single unifying "space program" in the US. (Unlike the USSR the US "military" space program was being attempted by ALL the branches initially with the main players ending up being the Air Force, Army and Navy in a distant third place) This would likely mean keeping the US space effort under the umbrella of an expanded ARPA (Advanced Research Projects Agency) though to be fair that's also contingent on who puts up the first satellite.
As for avoiding Von Braun and having Wiley Ley remain the main 'spokesperson' for the US one needs to keep in mind VB almost died several times during the war, (the "easiest" is probably have his car accident prior to capture being fatal instead of just 'almost' fatal) but his contribution OTL was initially being far more charismatic and persuasive than Ley or any of the other 'homegrown' Space flight advocates. No WVB and we likely don't see the "Colliers" series or the Disney adaptation there of and so the US is a lot less 'space minded' during the late 50s and probably even MORE shocked by Sputnik TTL.
Of course if you have no WVB then the US Army "Project Orbiter" (if it still happens TTL) is less likely to be opposed (as Ike can't complain about the "damn Nazi" in charge

) so you have a much better chance that the US will either orbit a satellite first, (same issues as OTL still apply so letting the Soviets go first is still possible) or doing so very soon after with a 'better' chance of success unlike OTL's "Vanguard" program.
Which in all means the US program could be slower due to the REAL "space race" rivalry between the USAF and the US Army taking funds from each other and slowing US progress.
Welcome to Alternate History
It may "feel" wrong but it's not actually that much of a stretch because of course 'factors' are different. As I noted above the 'original' purpose for creating "ARPA" was to coordinate the US military space program and try to draw together the various efforts but this was still rife with inter-service rivalry and other issues. The creation of NASA was specifically aimed at taking the MANNED effort away from the military while leaving most of the unmanned effort to the services so it's not really a stretch to see something like ARPA re-organized to fulfill the same function.
Randy