The battery is lead acid battery & its voltage will be 12-13.2
at nominal voltage (12V) the maximun flux in the core of your transformer using the push-pull topology is about 240mT. You can calculate this value using the following formula:
Bac_max=(V*10000)/(4*Ae*Fsw*N)=(12*1000)/(4*1,73*36000*2)=0.241
where:
V is the voltage applied to the primary in Volt
Ae is the the effective area of core in cm^2
Fsw is the frequency of switcing in Hz
N is the number of primary turn
Bac is the maximun ac flux in Tesla
he maximun value of flux depend on the type of ferrite, on the loss in the core , on the temperature of the core.
The shape of your transformer is ETD44, the type of ferrite I don't know, you should find something print on the core like 3C90, N67 tis is the type of ferrite.
The value of 240 mT should work but I would prefer to use a lower value of AC flux, this could be useful also in order to reduce the effect of the DC bias due to the problem of staircase saturation (flux walking) in push-pull.
To reduce the ac flux you can increase the primary turns for example 3+3 turns instead of 2+2 or increase the switching frequency.
If the average volt*second applied to the transformer when the first transistor is on is not equal to that applied when the second transistor is on, then there will be in the core of the transformer a DC component. More the DC current more the current switched by transistor and more the possibility to have the core saturated.
To limit this problem you could introduce a little gap in the core ( using the gap the magnetizing current is increased) or control the peak current, the TL494 has the possibility to do this.
Remove the diode 1n4148 I think isn't a good idea, this diode forces the pnp in off state when the gate of mosfet is driven.
To analize the waveform across the gate it is better to use a scope, a frequencymeter it is not suitable for this job.
If you want to increase the output current delivered from TL494 you should use a mosfet driver like a MIC4427.
Do you have tested the deadtime of the drivers?
stefano