If the memory slots have truly failed, I don't think there really is much to "upgrade" with a new motherboard, so you may as well replace it with the same model if possible, as that would at least ensure compatibility with what you have. It may be worth exploring whether Gigabyte offers any sort of manufacturer warranty that is still in effect, as I'd expect the memory slots failing would fall under it.
Now is a rather awful time to get a new Graphics card, as manufacturing has not kept up with demand. Even used graphics cards fetch a high sum- a GTX 970 for example goes for over $300 used. a 1070 goes on eBay used for nearly $200 more than I paid for mine new.
This makes it somewhat difficult to recommend any sort of upgrade. However it's possible the issues you've seen aren't related to your graphics card so it might be premature to explore a GPU upgrade, particularly in light of the excess prices of Graphics Cards. VintronNZ has an alternative explanation that might help.
otherwise, I'd suggest exploring your options within the 9 and 10 series of Nvidia Graphics Cards and AMD's R9 and possibly R8 cards and see how they compare to one another as well as what you have. If you google "Card A Vs Card B" you'll often get a result to UserBenchmark and GPUBoss websites which can give a reasonably good idea of how the two cards compare. It should also indicate aspects such as Power Draw which you will want to consider in cnoncert with your PSU's rated wattage.
