Like bert says:
We do not know your mechanical limitations but;
First you need to inspect the head as described or if not a mechanical tinker/piddler and repair stuff yourself it's a trip to the repair shop with the engine still on the machine so a good small engine mechanic can test. (lots of you tube info about your engine as to how to do this)
If you can remove and replace stuff yourself and find you have a bad head you will need a complete replacement head and a new head gasket and do the proper torque sequence. If the cooling fins/shrouds are clogged it will do it again.
New heads are pricy and complete used heads are more reasonable on fleece bay.
Them Briggs 28, 31, 33 series engines are prone to all kinds of failures, such as what you are seeing plus camshaft failures. Little bit of TLC such as keeping the cooling fin shrouds clean helps but even new engine can bite the dust at any moment. Cleaning the cooling fins, deck and complete tractor only requires about 5 min's TLC after each dirty condition of run time using a air compressor or leaf blower. I tell some guys this TLC stuff and it just seems to go into one ear and out the other and when I see a repeat customer ignoring such I tell them to go buy new. (their neglect of TLC maintenance is mainly why I have lots of salvage stuff for parts, but in your case a new Briggs engine of your model can flame at any moment, but little bit of TLC is also a good thing.
Also no need in checking with JD for Briggs parts. (the parts will be quite expensive over the JD parts counter, due to the grn/y colors)
Summary: You have to weight the odds yourself but installing a replacement engine on a worn out tractor is another thing to consider.
Don't hurt the messenger.