You can walk in short sleeved shirts. I have walked two
Camino Frances in late April and throughout May. However, you MUST have at least two long sleeve layers. For example, wearing a short sleeve merino wool or synthetic fiber shirt is okay IF you have a long sleeve fleece that can be worn under a long sleeve waterproof jacket.
I also recommend you include a proper head cap, like a synthetic "watch cap" or fisherman's beanie. Also include something to cover your neck. The ubiquitous Buff will do nicely. Finally, I always bring a pair of very lightweight microfiber "runner's" gloves for early morning and wet weather walking.
Winter survival training taught me to keep my feet, head, and hands dry and warm. If you are exerting yourself, torso temperature will take care of itself.
You WILL encounter snow at altitudes throughout the
Camino Frances in April and May. In two consecutive years, I encountered snow at Cruz de Ferro. Last year, O'Cebreiro was lost in a white out for more than an hour. It was not a lot of snow, but everyone had to shelter in place because you could not see the yellow arrows until the snow stopped.
I recommend the waterproof parka have arm pit zippers to help ventilate the excess perspiration and heat generated while walking.
Finally, on the stretch from Burgos to Leon or Astorga, temperatures can hit the high 'teens or low 20s, even in April, and certainly in May. This is one reason why I highly recommend zip-off trouser legs in addition to the short sleeve shirt. Make sure you have a good sun hat. A Buff, when wet and rolled, makes a fine "air conditioner" around your neck. I use mine that way all the time.
I hope this helps.