Hesitation and pulsing throttle

Experiencing severe hesitation when starting up from a stop and when accelerating while at lower speed. When engine is cold hesitation is minimal but as soon as engine reaches normal operating temperature. Hesitation is always so bad that I have to be careful pulling into traffic to avoid getting rear-ended. Engine never stalls completely and will eventually get going as long as you are patient and feather the throttle. Once hesitation is overcome and car begins to accelerate, acceleration is always in pulses. Even going up-hill with a steady foot on throttle just to maintain speed, acceleration is pulsing.

Have checked or recently replaced the following:
new MAF, new TPS, new fuel injectors, fuel pressure checked, good, new PCV. Engine new temperature sensor. Fuel rail was just replaced because of leaking pulsation dampener. New plugs, Denso OEM, throttle body cleaned.

No MIL or codes present.

Just recently noticed that if I stop and turn off car for short time (five to ten minutes), that when I start going again car runs like new (no hesitation or pulses) for a few miles but then symptoms quickly return.

Any thoughts as to what is causing these problems would be greatly appreciated!