Making sure my hash is right
So I'm digging something up from two years ago

01BTC10 corrected the hash, however, I wonder if he took into account it was a Dogecoin Core. I've been at it for a couple months now and I can't seem to figure out my old password. 

I ran bitcoin2john
And got the hash 

He corrected it to 

However, I wonder if that is correct as he didnt account for the fact its a Dogecoin Core key.

pycoin has the option to specify the network as Doge with 'ku -nDOGE' but I'm not sure what do do from here.
Why don't you just test it by generating a new wallet (with the old dogecoin software that you used) and trying to crack that specific "hash" (for which you know the password) ?
of course you should make backups of your original wallet etc beforehand
This is $bitcoin$96$5a757c74a47dd0f6550dc41ebd23ffc4d0cdb5f350037182db93e6feacd2e495cc87c94defdbb6fd7e2d149fbe08c493$16$4b128a9b40f33d07$265150$96$cfaa5b1ca5a706db612a72ffc2bf2d0a1086dce735bc8c3ba493f16a1f6860e853e5cf8cb3b549736fd50d8c7eda9a08$66$03ff36b83a7488e90edba467435de318d5c51fc75c1fd404b197c76a00158ba6c4:password

Session..........: 2018-01-05
Status...........: Cracked
Hash.Type........: Bitcoin/Litecoin wallet.dat
Hash.Target......: $bitcoin$96$5a757c74a47dd0f6550dc41ebd23ffc4d0cdb5f...8ba6c4
Time.Started.....: Fri Jan 05 13:27:46 2018 (1 sec)
Time.Estimated...: Fri Jan 05 13:27:47 2018 (0 secs)
Guess.Base.......: File (D:\HashcatGUI\dict\test-password.txt)
Guess.Queue......: 1/1 (100.00%)
Speed.Dev.#5.....: 0 H/s (0.11ms)
Recovered........: 1/1 (100.00%) Digests, 1/1 (100.00%) Salts
Progress.........: 3/3 (100.00%)
Rejected.........: 0/3 (0.00%)
Restore.Point....: 0/3 (0.00%)
Candidates.#5....: pass -> admin

Started: Fri Jan 05 13:27:39 2018
Stopped: Fri Jan 05 13:27:48 2018

However, the wallet.dat is from an older version, which is why I think the hash is off and had to be corrected.
Maybe the formating is what through me off. I got ku to show it as a format of Dogecoin and the results are the same. I guess its back to working on like I've been for the past two months.

PS H:\> ku --override-network DOGE 04c77ed36b8f0affb1b434ef6975b2fa224131f42ab47df30b7c6b6dffe47cc10c4d92b0bbb8205521bd7

input                         : 04c77ed36b8f0affb1b434ef6975b2fa224131f42ab47df30b7c6b6dffe47cc10c\
network                       : Dogecoin mainnet
netcode                       : DOGE
public pair x                 : 90234338766489736260560541387516194814900220115552850975862447215427335274764
public pair y                 : 35087268780626980418258611515549367557368934105453932834202777821135781664377
 x as hex                     : c77ed36b8f0affb1b434ef6975b2fa224131f42ab47df30b7c6b6dffe47cc10c
 y as hex                     : 4d92b0bbb8205521bd7b020db080b355acb75cf9ec04ebef0d2c3889f7369679
y parity                      : odd
key pair as sec               : 03c77ed36b8f0affb1b434ef6975b2fa224131f42ab47df30b7c6b6dffe47cc10c
 uncompressed                 : 04c77ed36b8f0affb1b434ef6975b2fa224131f42ab47df30b7c6b6dffe47cc10c\
hash160                       : a3759ac93719b9a3ce91d57413af6f7fbc96a215
 uncompressed                 : e53d032e405ce162fa75d054c453419051f053da
Dogecoin address              : DL3PZqfoZrMwBXwEAbgATryAN6HBxgqLWt
Dogecoin address uncompressed : DS3CNQTGgTCueQUJVFk2cAtLatpyHDyLvu
again, you do not need to guess.
The only way to be 100% sure if the hash is formatted/converted correctly is to use the exact same version of the software you used some years ago (the old wallet software, not a newer version of it) and generate a key with it.
After that you encrypt that wallet and you will see that the same problem happens with hashcat, i.e. that the public key/address is using the uncompressed format while hashcat expects the new one. Therefore you convert this test hash to the compressed version (you can also use ku for this) and try to crack this test "hash" that was generated with the same software version as your original "hash"/wallet.
If it successfully cracks after converting it to the compressed format, the hash that you are targetting to crack (not the test hash) will also be crackable for sure (assuming that you test the correct password candidates).
Thanks philsmd, I'll go back through and see which version of the wallet I had originally used and do that test and compare.