The other benefit of running and strength training on the same day is that doubling up encourages you to use a lighter weight and reduce the number of sets and repetitions because your legs are already fatigued from a run.
If the strength routine you’re trying leaves you trashed, adapt it so it can work better for you. Start with fewer exercises. Squats and lunges are great all purpose lower body exercises, so begin with those. Lighten the weight and reduce the number of sets and repetitions you did in the last workout.
You can even use your own body weight first, no dumbbells, and see how that feels. Focus on using correct form and going through the complete range of motion intended for these exercises rather than the amount of weight used. Add weight gradually and/or increase the number of repetitions over time.
Always allow yourself adaptation time for these new exercises before increasing the intensity with weight or reps, just as you would when increasing mileage. Adaptation varies among individuals, but expect it to take three to six weeks. It may take longer, depending upon how many kilometres a week you are running.
When you feel like you have adapted to the exercises, add in more. Also keep track of the amount of weight and the number of sets and repetitions you do. Increase the repetitions or the weight, but not both at the same time.
Keep in mind that running is your priority at this time, not weight training, and proceed cautiously.