I've been playing this game since it released. I bought the big box version and both expansions each on the day of release. I just purchased a new laptop. I created a gog account for the sole purpose to play the games on windows 10. Yet the graphics are abysmal. Inside the game settings you can choose either high or low graphics fidelity. The game is perpetually on the low fidelity despite clicking on high. Is there anyway to play the game with the better graphics?

GOG\Steam\Zoom any else digital distribution =/= Win10 support by default.

I assume you use GOG Galaxy to install and play. Do verify game files, at least twice in a row (cause GOG always redownload something).

BEFORE that - set game exe and\or galaxy exe to be run-as-admin (if galaxy set to admin, any games started from it also started as admin). If you install it in prgram Files - reinstall to X:\Games\GOG\Galaxy and game(s) to X:\Games\.
Even if you download, install, launch very exe as admin, still games shouldnt be in PF or on System drive at all.

*(game still be DRM-free and can be launched offline and withut GOG Galaxy, also you can download via normal browser or Galaxy normal offline installers, they still be installed to Games but you dont need to verify downloadings)

Go to WinRegistry and remove HoMM3 entries. Keep in mind, if you didnt start it as admin - even from games folder - you need use search to find that entries.
Or look carefully at PCGW articles - they can advise you where to search.
http://www.pcgamingwiki.com/wiki/Heroes_of_Might_and_Magic_III

After that better show screen where you change settings and what result you get.

There is no any pre-launchers settings for HoMM3 for PC-Win version (only for PC-Mac, but it not avaialable digitally anymore and never on GOG).

Additionally i can advise (in trying to guess what you actually mean here) - go to video drivers control panel and set stetching screen to "Keep Aspect ratio". This is neccesary for all games.

The in-game setting is for the cut-scenes.
For changing resolution I recommend the HD-Mod.

Or HeroesGL mod less features = closer to original. yet for screen featires - its better.

So it better show screenshots where you set high and what yu get. If that be not a bug but just GFX as it always be, than you need some mods.

GL wrapper Heroes3GL - best to start.
HD+ mod - next step to more improvments but less authentic than GL wrapper-mod.

if you comapre to what you saw on twitch streams about game - it probably WoG or more likely HotA mods. HotA include HD+ mod, but have even more HD Edition changes than actual HD Edition remake from Ubisoft (better re-drawned ui elements). More changes than HD+ mod.