Yes, the work permit will help you live more comfortably. But securing a place to live will be contingent on what income you have at the moment, which is only yours.
Where are you looking for places? Most of the apartment complexes with a functional website are listed as "luxury apartments" and will be priced out of your range. If you look through Craigslist you will find more options that are affordable to you. And if you're a Reddit user there's a sub /r/bostonhousing. In particular, the Somerville area is close to the red line and is densely populated with housing so you'll probably find something more reasonable there. Watch out for scams though, if anybody says they require a deposit before even showing you a place they're lying.
One awful thing I forgot to mention; nearly all Boston housing is run through real estate brokers, which means you pay an extra fee for somebody to print your lease for you. Often this fee is a full month's rent, but sometimes you can find places that will pay for half of it. For both California and Boston, consider that you will need to have at least 3 month's worth of rent ready to give up when you sign a lease. First/last month of rent and a security deposit. For Boston add another month for broker's fee, but maybe 1/2 month.
Stanford PI is correct, housing can be an issue. Again, Craigslist is probably a good option to find places to live. You can afford to spend a bit more on monthly rent due to the climate. In Boston, expect your winter heating bill to be over $100 each month. Cooling in the summer can be similar. In California it will stay fairly constant, maybe $40-50 each month, summer and winter. But you'll probably need a car in Palo Alto, you can live without one in Boston, especially if you're in an area near the subway.