I researched it for a similar time frame in '23 but ended up walking in July. My research basically showed the same as yours: weather can be very variable.
As far is the Hospitales is concerned the best advice seems to be always just to check the day before with your Hospitalero. Regardless as to the time of year.
Typically Albergues open from mid-march onward, it's very easy to look at Gronze just to check.
Time frame is very individualistic depending on how far you are comfortable walking each day, as most do it in 12-14 days you'll have plenty of time. Whilst many walk to Grado on the first day, if you're even remotely unsure about how you will cope I would highly recommend you stop five k's earlier at Paladin. The cafe restaurant was good, I'm told the albergue is excellent.
There's an excellent thread on prolonging this Camino (with shorter stages):
Hi all! Started today from Oviedo to Escamplero. Now I know, why so many people say it's a bit of a challenge to come out of Oviedo. I got lost just once, but was rescued by a very friendly guy. A short, but beautiful walk by great weather conditions - a bit cloudy, no rain and some sunshine. The Albergue in Escamplero is nearly empty and the food in the attached restaurant (Tendejon) is really good. More tomorrow.
You might find this thread particularly relevant as Thomas started on April the 18th. Regardless it is a very worthwhile read, with lots of info.
Like many I walked the Verde after Lugo, highly recommended.