Your 7 speed freewheel is old tech and is a bit of an orphan too - adding cogs increased the unsupported length of axle leading to more bent axles. 7 speed freewheel is really beyond the limit for most riders, and yours has probably survived this long because of your reduced load.
For the cheapest - you can get a freewheel that has a 13 tooth small cog. For example https://harriscyclery.net/product/shimano-13-28-thread-on-7-speed-freewheel-2694.htm at $23 USD. You'll want a new chain too.
However that only gives you 1/14th higher gearing AND you loose some climbing gear too dropping from 34 to 28 tooth.
There is a "mega7" 7 speed freewheel which ranges from 11 to 34 tooth, but good luck finding one. http://www.sheldonbrown.com/mega7/ shows 11-13-15-18-21-24-34 as the tooth counts, and looks like this:
They're long discontinued, so Ebay would be the best source.
On the other hand, you're wanting to get a better bike then its time to consider stepping up to exactly what you want.
A bike build starts with the frame, wheels and bars. They dictate everything that comes afterwards.
Instead of a 26" wheel look at a 650b road wheel size, and a road frame that is a suitable size for your leg length.
You're as far to the short side of "average" as I am to the "tall" side of average, so I know your frustration on sizing, but from the opposite side. You have the advantage of fitting onto a young-adult's frame which doesn't exist on my end of the sizing spectrum.
I'm not making a product recommendation here, but as an indicative example something like https://www.specialized.com/us/en/kids/bikes/allezjunior/128824 may be a good fit for you.
Its a 650 wheel, and has 8 speeds. You're unlikely to find 10 or 11 speed bikes in the smaller size wheels because there's just that much less room for the rest of the wheel's hub. Has the same dual-pivot brakes a larger road bike would have, and the cassette is 11-32 with 8 gears and proper road bike brifters.
In terms of gearing, 34/46 with 11-32 on 650 wheels gives you a lowest gear of 26.2 gear-inches and a high gear of 103.3 gear-inches.
That's pretty jolly close to my 700c wheel triple crank road bike with 24.5 gear inches to 101.1 gear inches.
IE its a full-on road bike with all the road bike features, and not a modged-about MTB.
I'd strongly stick with a decent brand name kids bike, because cheaper bikes are slightly more likely to be BSOs where the kid grows out of them rather than wearing them out.
Your final option is to replace the rear wheel hub with a 8/9 speed cassette-based hub. Shopping list of parts
- New hub with a cassette freehub. Number of spokeholes has to match existing rim.
- Matching cassette, 11-34 tooth 8 or 9 cogs.
- New right-hand shifter with matching 8 or 9 gears.
- New chain of 8/9 gears to match
- New gear cable inner and outers - they're cheap and may as well do it now.
You may require
- New rear derailleur, if your existing one can't move quite far enough to get to the top and bottom cogs.
- New spokes if the old hub and new hub are different dimensions
- New rim if the rear hub has a different number of spoke holes etc.
It is frequently cheaper to buy a new wheel complete than to try and rebuild an old rim onto a new hub. An 8/9 speed 26" MTB wheel should be relatively common to buy complete.
You should be able to re-use
- Existing chainrings provided newer thinner chain fits
- Front derailler, but thinner chain will shift a bit worse. You may need to overshift and then trim back, or maybe compress the FD cage a bit for the thinner chain.
Downside, this will leave you with a MTB frame with more gears, still heavier than a road bike. Plus braking performance may suffer using the MTB wheel brakes with dropbar brake levers. Really, not recommended for both price and performance reasons.