half bridge converter weird primary current waveform
Yes, your instinct is correct: that is a fairly typical drain current waveform when your core is saturating.

Your MOSFETs are not delayed, they are turning on when the gate voltage shows they should.  You can see the current ramping up slowly when the FET is on, while it says flat when its off.  Unfortunately, based on the waveform, your transformer is very poorly suited for your requirements.  

You have way too much winding inductance (too many turns) for your core...too many turns on too high of a permeability core. There is so much inductance that the current is rising glacially slow, and even that trickle of current is enough to make the nonlinear nature of the core be almost immediately apparent, distorting even the pre-saturation part of the wave slightly due to the more minor roll off of inductance before the ferrite just loses it.  Then, being ferrite, it begins to saturate like a brick wall as the small amount of current quickly overwhelms the core.

Transformers at higher frequencies can be somewhat forgiving when it comes to saturation as long as it is not too severe.  They can remain functional, more or less, but you're introducing significant high frequency harmonics onto the secondary due to the distorted waveform, which can cause all sorts of problems and cost efficiency.  A simple resistive load is going to tolerate the harmonics the best, but other types of loads, or larger ones, will probably make your efficiency plummet.  

Anyway, transformer saturation is sort of wily in that things can appear to be working (and are, at least if you don't care about EMI, harmonic losses, efficiency, overheating) for many cases of core saturation.  If you saturate an inductor in a DC/DC converter, usually something is kind enough to explode or at least catch fire, alerting you to the problem quickly and obviously.  Transformers are not so nice and like to hide their problems and act like everything is fine, even if its not.  

I would try reducing the number of turns while preserving the ratio.  I know you need a lot of inductance to handle that much voltage no load, but you have too much even then.  At 30KHz, the MOSFETs aren't even able to ramp the current up much at all.  You should be ok to reduce the turns a bit.  Just remember, inductance is proportional to the square of the turns, so don't over do it.  On the plus, if you over do it, it will probably melt or catch fire, and then you'll know (unlike saturation, sneaky bugger).