I used to work nights and had the same issues at times. Switching to day shift will most likely help, but weight lifting on days off worked well for me.
I think night work can change the entire rhythm of the body. And actually with regards to erectile dysfunction, they've looked into it, studied it and yes, erectile dysfunction IS more common if you are a night shift worker. https://medivizor.com/blog/SampleLibrary/erectile-dysfunction/do-shift-workers-have-an-increased-risk-of-developing-erectile-dysfunction/
Over and over again, it is said that shift work will hamper you in this way.
Is it at all possible to work toward a day job only?
Morning erections are not a sign of anything. Mostly we get them when we're young and then we don't get them much anymore, although obviously people do differ. So don't worry about those, they're irrelevant. But yeah, if you're tired all the time, you might have trouble doing all sorts of things, and erections might be one of them, although there's certainly no guarantee this is so. Shift workers have a problem with their circadian rhythms often, which affects the hormone melatonin which sets our sleep cycle -- we don't all have the exact same cycle, but everyone has a normal cycle -- and melatonin is made from serotonin, so it's possible that some shift workers will get neurotransmitter problems that might leave them a little on the down side emotionally, maybe dulling their senses a bit and that could also be a possible factor. Testosterone deficiency is less likely if you're relatively young, and I'm assuming you are as you're still concerned about those morning erections. Keep in mind, the main reason for morning erections is that's when for most people, though again individuals do vary, that's when testosterone levels are highest, but for shift workers when exactly is morning? And if you're burning the candle at both ends, working both nights and days, then when is it exactly you do sleep? You can go get your levels tested, and they should have been tested at your last physical, but to truly measure this you have to get tested over a period of time, not just once when you might have just been at a low point in the day or a low day that isn't your norm. But if you're thinking of leaving the job anyway, I'd just suggest that you see what happens when that happens, assuming you're planning on leaving soon. Peace.
It is certainly possible that your physically demanding job, just from being that physically demanding, has made you less likely to get an erection in the morning. Have you had your testosterone level checked?