Small local hospital for dd1 and dd2- the LDR rooms were all private but the postpartum rooms were technically "doubles". They had 6 postpartum rooms and rarely needed to double mothers up, but they did let you know it was possible and if they did double you up then your partner could not stay overnight.
Larger hospital planned for this next babe- again they have private LDR suites but it's a newer/larger facility and all of their postpartum rooms are private.
If your insurance is causing problems, ask the hospital billing office for assistance. Good luck!