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Third party parts will usually give you third party problems Try this iPhone 5s Lightning Connector and Headphone Jack Use this, just in case iPhone 5s Lightning Connector Replacement
I’ve heard of heating the Baseband chip for 4-5min at 400f and it fixing it. but first try backing up and restoring. if it throws error -1 then try reflowing the chip.
Im not 100% sure about the battery issue you are having, but check the time on your phone. I always unclip the battery when I do screen repairs and I had the searching issue on an iPhone 5s. I checked the time and it was set to 1977.
