Actually finding dead fish stuck in what is definitely hair algae suggests one possible cause for the issue...too much organic waste in the tank for the algae to feed off.
(The moss balls don't send out hair-like threads of themselves to reproduce. The moss-ball algae is very different).
So...suggestions for control;
Less light...duration, not intensity or colour. (Algae can usually perform better in some spectrums).
Less organic waste.
Less plant fertiliser.
Get yourself a bottle brush, or similar, such as a toothbrush, to twizzle the hair algae out of the tank, best you can.
DON'T be fooled into thinking you can buy an 'algae eater' fish to sort out the problem...very few will touch hair algae and would most likely ignore it whilst there's other stuff to eat. Snails seem to ignore it as well.
Seconding all of the above!
My two year old tank here often gets hair algae since I started it two years ago;
The only way to beat it is to stay on top of manual removal and tank maintenance, and to resolve the underlying imbalance in your tank that is causing it to thrive.
It is NOT caused by moss balls.
Otos, mollies etc don't eat hair algae, and most fish won't eat problem algae.
A bottle brush is the easiest way to manually remove it, as Bruce recommended! I use a large one to search for and check for wisps of it, and a small brush to get down among the tall and thick plants and manually remove any, before it gets thick enough to pose any threat. The bottle brushes are also great for cleaning filters and tubing.
In my experience with it in this tank and the garden pond - hair algae isn't usually sticky, and doesn't pose much threat to fish unless it's been allowed to get out of control and has become very dense. Then those fine strands can potentially get wrapped around fins or gills, trapping a fish. But the only time I had that happen was as I was manually pulling a dense spot of it out from behind a plant, and a baby pygmy cory had got caught in it
as I pulled the mass of algae out. I'm positive that it must have only become trapped by my pulling it out because once I gently freed it, it was absolutely fine, so can't have been caught there for very long. I've also seen the photos and experience
@CassCats shared here where a tiny fry had accidentally got wrapped up in a strand of christmas moss - so freak accidents like that can happen with any fine plant too (her baby lived as well!)
I'd suspect that your dead fish (I'm sorry for your losses) may have just hidden among the plants/algae while dying, rather than dying as a result of being trapped in it. But that is just a guess, and I admit that it could happen as a freak accident - but two freak accidents makes me think that it's unlikely to be the cause of death.
That experience reinforced for me that it's essential to stay on top of tank maintenance with this stuff. This tank has a lot of botanicals and plant matter, so getting the light and nutrient balance right is a constant balancing act. Tank not cleaned and maintained often or well enough means lots of organic material breaking down, and algae thriving on those extra nutrients. Not enough ferts and the plants start to fail, taking up less nutrients themselves equals more nutrients for hair algae, and it starts reappearing again. My light on this tank is too bright and without floating plants, hair algae returns, so I know I need enough floating plant cover in here.
It does grow fast, but it thrives best and only poses a risk to fish if it's allowed to continue forming dense little areas, often in untouched or shaded areas. In my tank, that's behind the dense planting at the back, and attaching itself to the driftwood. Substrate cleaning is essential, because decaying plant leaves and botanicals leave organic waste that algae seems to thrive on if there's too much for the tank to handle.
So weekly maintenance has to include a sweep with a bottle brush, and manually removing any I find. It's an easy quick task in here now and just part of the routine, provided the balance of light/ferts/organics is right.