Yes, I'm sure they'll handle both. Free clinics can provide a great base to launch from. They're also great for linking in with other specialist services like social workers etc.They are trauma specialists also, but only specialize in sexual traumas, I not only have sexual traumas, but also bullying traumas, so I am not sure whether they would help me with multiple types of traumas.
Complex trauma, unfortunately, tends to take protracted treatment. Free clinics may be able to provide you with ongoing support, particularly crisis support. So they're very helpful in that regard. But they often have their limitations for chronic conditions. See how it plays out. If they can't provide comprehensive enough care, they should be able to help direct you to somewhere that can.
Your GP is also worth speaking to. They've probably already done a plan for you to get psychologist support. If that's not cutting it, you may want to talk to them about a (trauma-focused) psychiatrist referral.