try doing your warm-up and then when you lift, do more reps (10-12) and/or more sets(4). make sure you're still keeping good form and not cheating to finish out the last few. you'll have to adjust your weight settings. also, do a circiut to work the whole body.
then go into your cardio. if this is still too much and you feel like giving up, alternate your days, but do more cardio(running) for the stamina.
when you incorporate the weight training you'll be building the muscle you need for the endurence.
remember, if you want to keep weight training, and you should, make sure your form is good and you aren't cheating. it's quality, not quantity. and if you're too tired to lift after your cardio, don't sacrifice the form.