The small frame point is oft misunderstood. I've had countless people telling me I need a bigger frame because of the amount of seatpost I require but nobody saying that ever takes the triangle into account. When a dimension is lengthened the other two (seat tube and downtube) are also affected.
I've had identical bikes in different sizes where the difference in actual TT length is less than 10mm, but when saddle height and fore/aft are set appropriately, reach increases significantly on the bigger frame size. Conventional internet "wisdom" would suggest a shorter stem, but this in reality is misplaced as the physical frame length hasn't changed
Bingo!
You have to consider the relative positions of the 3 contact points, all else is mostly irrelevant. If you're using muscles to keep position on the bike you're wasting energy.
I have the similar problem to the OP but for the opposite reason. I am short of leg and long of back and only 5'9". On a 'standard' sized bike properly 'sized and set-up' I always end-up pitching my weight forward onto the bars and constantly need to shift my weight back on the saddle just as the OP describes. This is due to my centre of gravity relative to the contact points which I need to move rearwards to counterbalance my long body.
In his excellent (I think the best) article on bike set-up, this idea of being balanced on a bicycle is well described (forget the crank length nonsense).
https://www.peterwhitecycles.com/fitting.php
If you stand straight, bend at the waist and start to reach forward your arse moves rearwards as a counterbalance over your feet. This is how you should be on a bicycle* at such a seated position over the pedals that you can easily rest your fingertips on the bars with your torso counterbalanced.
I found on standard bicycles that despite fitting the most set-back post I could find I still needed to sit on the rear of the saddle to find the optimal position for pedaling and all-day comfort.
In the end I bit the bullet and went for a custom frame in order to get a more laid-back seat-tube angle as my primary goal. Unfortunately, this was not clearly understood and I ended-up with a lovely frame, but the same problem. So I went back to the frame-builders and had a long and at fitrst mildly heated discussion with Brian Rourke about what I wanted and why, and we he started to talk in terms of being balanced so that you can simply rest your fingertips on the bars THEN we understood each other. I ended-up taking a saddle on a set-back seatpost on a broom handle to show him where my seat needed to be relative to the Bottom bracket compared to the bike they built and HOORAH, I have a bike with a longer than average top-tube BUT where I sit perfectly on the saddle behind the BB to simply touch the bars with my fingertips.
The best fitting bike I ever owned apart from my 1950's Holdsworth made in a time when seat-tubes were very laid back and riding styles were different even when racing.
OK, long-winded experience, but part of this is that not only does raising the saddle lengthen your effective leg length/knee bend so does moving the seat backwards and both give a different sense of balance/comfort/efficiency.
The cheapest and most set-back post I could find at the time was this one, the same as the Velo Orange version and designed to work with Brooks saddles, but works brilliantly with all regular saddles. If the post length is long enough it's a cheap option to try....
https://www.planetx.co.uk/i/q/SPHOGS/holdsworth-gran-sport-seatpost