Here is a link to view parts for your engine.
Briggs and Stratton 288707-0170-01 Parts Diagram for Carburetor (jackssmallengines.com)
Look at the item 125 carb at this link and when you look towards the bottom of the page it will show several models Briggs series 28's and other makes of machines that used the Walbro LMT carb. at this link
Briggs and Stratton 799728 Carburetor - Jacks Small Engines
You will most likely end up getting a $15 Nikki carb so as to test the engine, since you have never seen the old L-head type engine run. (try not to throw good money into something bad)
(and you will find that one size does not fit all with the Nikki replacements)
What I'm indicating is internally inside the carb unless you buy two at a time and then you can play with enlarging the jets if the engine runs lean if both carbs are exactly the same. Take two carbs and make one. You might get lucky and get a good run first time with the Nikki, but I'm usually not that lucky when playing around with Nikki carb replacements.
In meantime you can keep a eye open for a 28 old model 28 series Briggs engine that has a Walbro carb. (The Walbro carbs are more user friendly for myself and my engines vs playing with the Nikki's)
Quite common for a Nikki carb on a 28 series Briggs engine to surge and hunt, especially when the engine is cold. Adding a restrictor plate will sometimes help reduce surging (and some Nikki replacements come with a restrictor plate gasket or adding little bit of choke, but yours has the choke plate on the throttle linkage a full open throttle.
As Star Tech relates calling the current Briggs place is a waste of time and effort. You can get better hands on info at this site for problems such as yours because been there, done that.
Here is a good reference link for small engine carb repairs. (all kinds of small engine carbs)
https://outdoorpowerinfo.com/repairs/