My internet connection at home is still wonky, so I'll be brief. You should look for a special-purpose radio for your storm room. Get one of those combo crank-up/solar/AC powered AM/FM radios. I used mine when the power went out for three hours last week during a storm. If you gotta bug out, take that radio with you, along with your bugout kit. There are various brands on the market. Some also have short wave bands.
Some FRS/GMRS radios may have weather bands on them, as do some portables and scanners. I just don't think there's one single device that will do all that you want, especially under $100. But you might get away with two, which also gives you some redundancy.
Ok. I'll keep an eye out. One of those crank ones would be great! Even if it's not a storm alert, I would still have the radio...that's what I need the most...a radio.