Handlebar width?

Try dropping the front tyre pressure a bit.

I find this lengthens the contact patch a bit and hence reduces how keen the front wheel is to turn. Sounds a bit odd but it works.

I don't find width makes that much difference, but 38cm sounds very narrow for a cross bike. Most cross bikes are fitted with 42-44 cm width bars.

I guess the width is body / frame size dependent to a degree. I am six foot, my 56cm cross bike has 44 cm bars (2cm wider than my road bikes).

Go as wide as you like, get some 46 cm shallow drops if it makes you comfy, I use 44's at the moment and that's only because they didn't have the 46's I wanted in stock at the time.

You don't have to ride on 38's just because someone tells you to, It is not compulsory to give your self a humped back through always using the tiniest bars possible, you are not in the wind tunnel or setting the hour record.

One way to stabilise the handling a bit is a longer stem, however, that will obviously affect the reach, might not be doable if your fit is already on the large end of the spectrum.

Having said that, when I transitioned from my flat bar hybrid to my first drop bar bike I was really, *really* taken aback by the difference in handling. I too thought it seemed twitchy, less controlled. After a few weeks that changed and started seeming much more natural. A short time later I was completely used to it, and I found wide flat bars just as weird.

So it might just be a matter of giving it more time.

You might also want to consider a flared drop handlebar, like the On-One Midge. It's recommended that you do raise the bars if you do so, as you'll spend much more of your time on the drops. You should check out pics of them mounted on bikes to get an idea of what they are like, as the unusual angles are difficult to imagine from a shot of the handlebar by itself (at least, that's what I found).

The width increase will make slight diff, also many pro cxer's run wider bars.
Its most likely something you could get used to but I would get some cheap 42-44cm bars and give them a try.
Bringing the stem up slightly could help with weight distribution but this is Dependant on your current fit.
Hope you're healed up after your crash, there's not much that can done once you've started slipping on diesel....