Hola and welcome.
A difficult question to answer as trying to predict the weather from our firesides is almost impossible. The only real way is to check the on-line forecast at
http://www.aemet.es/es/eltiempo/prediccion/municipios and use the interactive map to check for conditions. The snow line needs to be above 2000mts to get over some passes, and it is lower all this week. Listen to the locals who will know if the passes are open and check for the next night's stop before you set off. If you need transport you need to be aware that there is no bus up from Oviedo to Grandas on a Saturday, and only one per day late afternoon on other days. I suggest that you check the ALSA site. Between Grandas de Salime there is no public transport of any sort. In really bad weather the passes can be closed to heavy traffic - the
Asturias112 Estado de carreteras is a good site with a roll over interactive map. I have never been able to find one for Galicia, but you are working downwards after Alto de Acebo so it isn't so important. Do make sure that you have a phone and taxi numbers for emergency.
Having said that the weather could be mild and dry and walking easy.
Gronze should have up to date information about albergues.
Hope the weather is kind to you
Buen Camino