Have to see what is the total load including the dead load and the live load on that deck.
Dead load is basically the load or the weight of the deck itself. In some cases its usually 10 DL. Then you got the live load which is created by all the extras you got on the deck, like furniture, flower pots, and people which is usually per code is about 40 LL so together the design load is 50 LB PSF. Now if you put a hot tub on the deck and in the winter you get 4' of wet snow, etc the load could increase to
60-80 LB PSF, so even the 2x10 which is being spanned to its max, the current load conditions on that deck could play a big part to cause the bounce, deflection, etc. So if the joists sag as you said they do, something is not right there and you should consider breaking that span up to reduce the span so it can support the load which is placed on the joist that is causing the sag... because if you leave it the way it is, and people will have more people then usual on the deck and they start to jump around, that deck could collapse.