I think the choice of 20W-50 was a mistake for a number of reasons. I would not try additives.
I had much the same problem for quite a while. I finally tried synthetic Mobil1 0W-40. Not had a lash adjuster problem since, even in cold weather (North Carolina). At the time I put the Mobil1 in I had 123K on engine with original heads and bottom end. I only put about 5K to 6K on this truck a year so I'm doing once a year oil changes with a new filter and top up at six months. This helps offset the cost of synthetic. I'm not advocating Mobil1 as probably any 0W-40 grade would work. But for me Mobil1 is easy to get and about $4 per quart which is about half of what a cafe' oil would cost. I use Redline on my 03 Ford Ranger and Castrol non-synthetic 10W-30 on my 95 Camry. You may also want to change the oil at 1000K with the switch to flush out all of the 20W-50.