The SWAN all-sky camera on the Solar and Heliospheric Observatory (SOHO) spacecraft detected the hydrogen Lyman-alpha (Lyα) comae of comets 2001 Q4 NEAT and 2002 T7 LINEAR for large portions of their perihelion apparitions in 2003 and 2004. C/2001 Q4 NEAT was observed from 2003 September 14 through 2004 November 2, covering heliocentric distances from 3.23 AU before perihelion to 2.75 AU after, and C/2002 T7 LINEAR was observed from 2003 December 4 through 2004 August 6, covering heliocentric distances from 2.52 AU before perihelion to 2.09 AU after. We combined the full set of comet specific and full-sky observations and used our time-resolved model (TRM), which enables us to extract continuous values of the daily-average value of the water production rate throughout most of this entire period. The average power-law fit to the production rate variation of C/2001 Q4 NEAT with heliocentric distance, r, gives 3.5 × 10²⁹ r ^–1.7 and that for C/2002 T7 LINEAR gives 4.6 × 10²⁹ r ^–2.0. Both comets show roughly a factor of 2 asymmetry in activity about perihelion, being more active before perihelion. C/2001 Q4 NEAT showed a production rate outburst about 30 days before perihelion (2004 April 15) and then a large extended increase above the nominal trend from 50 to 70 days after perihelion (2004 July 5-July 25).