"is New England really going to go into the season with Jarrett Stidham as its starting QB?"

I believe that is the plan. I've read reports New England currently has about $25 in cap space, though they will probably move some pieces around because they're going to have to set aside money for their draft picks, but unless they gut their roster there's no way they're going to clear enough space to upgrade the position significantly. 
For now, the "In Bill We Trust" mantra is holding, but if the Patriots start slowly, the mutinous forces are going to gather quickly. I have no doubt Goodell will give them a couple of layups to start the season - like the Jets and the Dolphins at home - but their schedule outside the division isn't easy. They play Kansas City and Houston on the road, they play Baltimore and San Francisco in Foxboro, and they have to travel acorss the country three times to play both Los Angeles teams and in Seattle. Even though I loathe the 49ers, my dream scenario is Garoppolo lighting up the Patriots for about 400 yards and 5 TDs in front of their feckless fans.

APIAD wrote:

They about have to play Tampa correct?

Not unless they meet in the Super Bowl. The AFC East plays the AFC West and the NFC West. 

https://fbschedules.com/nfl/new-england-patriots/

In 2008, New England also had the two West divisions in their schedule rotation. Goodell arranged their schedule so they played at San Francisco and at San Diego on consecutive weeks, and then at Seattle and at Oakland on consecutive weeks. They stayed on the West Coast in between games both times. So instead of having to fly across the country four times, they only had to fly across the country twice.