Two major issues. The first is cadence and beocming winded; the approach here is to improve conditioning and to reduce the gear ratios. Bigger cogs with smaller chainwheels, as has been noted.
Second one is front-rear weight distrubution. The classic frame design guides (at least Talbot, Oliver, and CONI) plus the modern Arnie Baker say there should be 40 to 45% of the bike+rider weight on the front wheel. Saddle fore/aft positioning affects this, as do chainstry length, and front-center. I've found it's a lot more comfortable for me to get this on my Woodrup with a 44 cm chainstay and 60 cm front-center, than on my Mondonico with 41 cm chainstays.
Obviously leaning forward accomplishes something similar, but it's a lot more tiring. My HR soars faster if I stand.
I can't say this will work for you, but it should get the bike more working for you. It argues in favor of a sport-tour geometry rather than a modern road race geometry.