You must file a federal tax return regardless of your state of residence.
If your main, permanent home is in TX, and you are in MO temporarily in order to attend school, then for state tax purposes you remain a resident of TX. In tax terminology, TX is your domicile. TX, as I'm sure you know, has no state income tax. However, any income you earn in MO, including remote work done in MO for your employer in RI, is taxable by MO. (The remote work is NOT taxable by RI if you never physically work in RI.) ) Therefore you should request your employers to withhold MO taxes. You will file a non-resident MO tax return** at year's end to report your income.
**If you maintain a "permanent place of abode" (such as a rented apartment) in MO, and you spend more than 183 days of the tax year in MO, then you would file your MO return as a resident. Note that a college dormitory is NOT a "permanent place of abode" for tax purposes.
https://dor.mo.gov/pdf/nonres_flowchart.pdf
Where do you want your employer(s) to mail your W-2's next January? That's the address you should put on your W-4's. But make sure your employer(s) withhold MO taxes.
**Answers are correct to the best of my ability but do not constitute tax or legal advice.
