Certainly, you can do this. TurboTax can easily handle (2) part-year resident state returns. This will not pose any particular problem in the software, especially if your W-2s and 1099s already have the proper state information allocated in the relevant boxes found near the bottom of the respective forms.
What is most important, however, in completing a tax return of this type in TurboTax, is to "properly" set up the Personal Information section in the first instance -- so that the program will "know" to add part-year state returns for Connecticut and Washington, D.C. after the federal income tax return is completed.
A graphical example of how this should look is provided in the screen-capture images found directly beneath this text; simply click the pictures to open. While your information will be somewhat different, of course, to reflect your unique individual circumstances, the basic set-up will be the same.
Please pay particular attention to the drop-down menus regarding residency, move dates, and the rest of the biographical data. Once that has been done correctly on the federal side, and after inputting your W-2, 1099, and other taxpayer data, the program will then guide you through the (2) state filing process in the state modules, and ask any necessary questions about allocation of income, deductions, and credits.
You may also find the following TurboTax reading material relevant and helpful:
https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Planning-and-Checklists/Multiple-States---Where-T...
https://turbotax.intuit.com/tax-tools/tax-tips/Tax-Planning-and-Checklists/Multiple-States---Figurin...
Thank you for asking this important question.
