I.G.
Doulas are not medical professionals, their role is to support a woman before, during and after delivery. They do not do prenatal testing. and they not not provide medical prenatal care.
Midwifes may or may not offer prenatal screening like the first trimester screen and nowadays maybe even MaterniT21. They will usually refer you out for the Ultrasound though...
If you intend to have an invasive procedure like a CVS or an Amnio they will refer you to an OB.
If you just want prenatal care by a midwife but do not want her to be present for your birth it really depends on your circumstances whether or not they will be able to take you on as a patient. If you are high risk pregnant and must deliver in a hospital a midwife may decline you as they specialize in normal low risk pregnancies.
If you need prenatal care now, but maybe you plan on moving before delivery that is usually not a problem.
I would just call around a bit and ask...
Good luck.
Edited to add: If you want a natural birth in a hospital I would find a CMN (certified nurse midwife) with hospital rights that accepts your insurance and respects your wishes for minimally invasive prenatal care and birth. They exist! I had one and had a great experience throughout my pregnancy and birth. Your insurance company should have a list of preferred providers... start at A and call all the midwives on your list. Good luck!