If my husband and I live in different states (for work purposes) and therefore have different primary residences, can we still file "married, filing jointly"?

Yes; as married couple, you only have two requirements for filing a joint return. One is that you and your spouse both agree to file married filing jointly; if you don't, you have to file married filing separately. you can file a joint return when living apart, UNLESS you're legally separated. Since as you said, for work purposes , you and your spouse live in different states, you still need to file your state return as MFJ as your federal return filing status. I guess either you or your spouse is a nonresident of another state.

NOTE; this is not your case. However, on your state return, you May file either MFJ or MFS aslongas both of you have different residence status; for example say you are a full year resident of CA while your spouse is a full year resident of IL, then you can file MFJ or MFS on your CA return so can your spouse.