Best to buy a cheap non trunking or just a cheap used scanner specific for Weather Alerts if really a requirement.
Weather Priority is not something I would want to put up with while scanning.
Plus, realize that if you use the weather priority, it's only checking for the alert tone, so you'd get alerts for anything sent out from whichever NOAA station you set it for, some of which would likely not be of interest or importance to you, i.e. not a threat since the storm(s) may be too far away, not moving anywhere close to approaching your location, etc.
While you can enter the S.A.M.E. codes, which will show the specific alert for whichever counties you choose, that only works if you put the scanner in the correct mode for those to be used. If in that mode, then the scanner is only searching for the codes for the areas you entered. It cannot scan regular channels while listening for the S.A.M.E. alerts.
At home, I have a couple of weather radios, one in my bedroom, the other in my home office at the other end of the house. If I'm going to be out and about, away from home when there is a possibility of strong storms, I'll take one of the older, seldom used, scanners to set up as a 'weather radio' if needed.