Will my existing deck support the weight of a new hot tub?
Double 2x8's spanning 10' @ 16" O.C. can barely support 160 PSF (safely) if they are douglas fir and the spa is centered on the joist.  150 PSF over ~50 Sq. Ft. would be about 7,500 lb.

As for your end beams, (3) 2x10's spanning 7' would support the weight of the joist ends with a 1.5 sq. ft. footing on each end of the beams; 4 total.

Just to stiffen things up, I would add (2) 2x8s between the existing joist evenly spaced ~5" O.C or so to help support the deck boards.  With the beams and the footings, this would give plenty of support to the SPA.