I have taken ferrous sulphate and a different time, ferrous fumarate, with no issues, apart from poo going dark green/black. So I think whether your stomach will take it or not is pretty individual, and you won't know till you try.
I never seem to get my levels up consistently well, though, and currently it's stopping me from giving blood - barred for 6 months.
I've been tested for coeliac (runs in the family, but not me,) but I also read recently that Metformin can reduce it, which I am on for type 2 diabetes. So it might be worth checking if any meds you're on might have an effect, or if there could be conditions affecting absorption.
I've also had B12 pernicious anaemia, too in the past. I am a meat eater, including red meat and occasionally liver, and I est a fair few leafy green vegetables and so on. I've never drunk tea or coffee. I just seem to have crap blood as far as iron goes (because it predates the Metformin by a couple of decades.) I suspect mine was exacerbated by peri periods and their flooding glory, but I haven't had one for dome while. Obviously heavy periods do increase the likelihood of anaemia, but apparently anaemia also increases the likelihood of heavy periods, which doesn't actually make much sense to me, but there's bodies for you.
Good diet, regular exercise, etc, etc - but it can get hard to manage that if you're just knackered all the time.