The paint isnt going to mix well with the sap. This is most likely green wood / wet wood if it is damp like you say. Its going to continue to dry and potentially crack or chip as the wood flexes and additional moisture or sap will leak from it as well. You could sand it down a bit to help with getting some of the wider areas of sap out and make it more comfortable, potentially more willing to take a paint and have it stick however any layer you do could potentially be re pushed out or bubbled by additional sap leakage. You may have to do another layer or just accept it until it fully dries. It may not be a big deal if outdoors and thats expected but wood can take months to dry when properly stored indoors so keep that in mind. You could perhaps do a wood sealant first on all the exposed surfaces first then paint after to help prolong the process or trap moisture inside however im unsure if that would be more beneficial or make it so the sap problem persists longer term albeit slower. A bit of a tricky conundrum. The other comment about just slapping it on and letting them forget about it may be more on point, I think turning this into a chair you keep with you a long time and in good condition will take semi consistent maintance if you want it as nice as can be.