The short answer is that during El Nino fall winters and springs, the polar and tropical jet stream tends to track over Texas more often, leading to a higher amount of rain making upper level disturbances.
With La Nina the jet stream shifts north of Texas in an arch like configuration, keeping it warm and dry here for longer periods of time.
Got that part J. But do you know why the jet moves north or south? What sets up the high pressure and sinking air in the eastern Pacific.
This would be a fantastic topic for the Weather Talk category, and @Joseph I’d love to hear an answer, too!
I honestly don’t have a complete answer for that. I have wondered that myself for years but can never find a clear answer. My best guess for La Nina is that the cooler waters causes a stronger more broad high pressure area to develop because the cooler temperatures over the Eastern Pacific is more dense. Maybe this bigger more strong high pressure that develops is what nudges the jet stream to slightly higher latitudes.
With El Nino, you would have a persistent large area of low pressure over the Eastern Pacific because of the warmer rising air. The warmer air from the warmer Eastern Pacific may strengthen and position the jet stream a little further south than usual because the temperature contrast between the northern and equatorial Pacific is sharper than it is during La Nina. Jet streams generally are created by sharp temperature contrast after all.
This could explain why the jet stream winds track a bit lower during El Nino events. Texas just happens to be in that sweet spot where mid latitude cyclones hit directly more frequently.
Disclaimer: I am not at all saying this is 100% factually what is happening scientifically. This is just my best armchair meteorologist guess.
The inaugural topic for “WEATHER TALK”!
(Hoping now, Lee will work on team name El Niño)
Re the jet moving…I agree it’s the egg…and the chicken is the emerging trade winds blowing toward the western pacific (la nina) which cools the eastern SSTs. What I have never seen is any science on why for 4 out 5 last years these winds are stronger…just a random throw or another cycle? I guess smarter people will tell us. 
Yeah It does seem to be a random cycle with no distinct pattern to follow. You have probably seen me say this before in previous comments over the years: When it comes to the weather, flip a coin and that’s what you’ll get.
And personally I much prefer the El Nino fall/winters because I love stormy weather and cold dreary winter days, which we usually get much more of with El Nino, aka “the little boy”