Sounds like this may be a battery issue. A battery replacement is inexpensive and a good place to start.
Next step is indeed a motherboard replacement.
[https://www.amazon.com/dp/B00FBM4E5O/ref=sspa_dk_detail_0?psc=1&pd_rd_i=B00FBM4E5O&pf_rd_m=ATVPDKIKX0DER&pf_rd_p=f52e26da-1287-4616-824b-efc564ff75a4&pf_rd_r=VR3DHDTAG7K396F053B5&pd_rd_wg=wgLnU&pf_rd_s=desktop-dp-sims&pf_rd_t=40701&pd_rd_w=Z0Z4N&pf_rd_i=desktop-dp-sims&pd_rd_r=43b36524-da4c-11e8-812b-6f7e56fb9f31|PSP 2000/3000 replacement battery]
[guide|1267]