Should I Take Peri-WO Protein and Carbs if I Lift/Run in Same Session?

First I’ll state that my current goals are to improve body composition (mainly lose body fat, muscle gain would be a bonus) and improve my conditioning. I will be doing a full body strength workout 3 days per week that consists of 25-30 minutes of strength work (focusing on deadlifts, front squats, incline pressing, overhead presses, pull-ups, rows, dips) followed by a 10-30 minute conditioning session (sprinting, farmers walk, and/or 2-3 mile tempo run). I’m combining strength with conditioning/running mainly to be more efficient with my schedule. The question is, should I consume some protein and simple carbohydrates after my lifting and before conditioning/running or is it not necessary? thanks,

Yep, you should have workout nutrition before and during the lifting, and probably during the conditioning too.

The right nutrition, timed properly, can help performance during the session and get a jumpstart on recovery.