As always I recommend seeing a specialist, however from this base info...
It seems like you have a ANTERIOR PELVIC TILT Distortion and maybe even a LATERAL PELVIC TILT Distortion. I find also when this occurs it can also mean there are lower postural distortions as well being that it all creates and contributes to balance or lack there of.
The good news is that even though chronic these distortions could be easily managed or even repaired with the right program. The key though is to treat it no different then you would your lifting routine as far as planning and designing your program. Basically make its importance a priority.
I would do the following...
-Really focus on dynamic warm ups and specific warm ups pre-lifting, especially when doing any lower body work. Also I agree with your idea of unilateral work particularly contralateral DB Deadlifts and 1 Legged Squats.
-Design a plan for 5-6 days a week where you have an intensive repair session. Rotate days between SMR work with foam rollers and static stretches. If can include some band work with those static stretches as well.
-Post lifting go right into your stretch program (with re-feed shake/meal in hand of course

This is the most crucial time for repair, plus your muscles are nice and warmed and prime for stretch and deep tissue work.