I don't think that is your particular problem but maverick44 makes a VERY good point about maintaining operating temp. A cold running engine starts having "health" issues in short order along with inordinate fuel consumption.
Yes, you can do an "in frame" overhaul on a Detroit with the block in place because of the modular construction of the engine. That is, of course, if you have, or can gain, enough clearance under the block to access the rod caps. The rebuild is done with what are called "liner kits" where new cylinder liners with matched pistons are installed to, essentially, form a "new" cylinder in each hole. I would not recommend just throwing new rings at it because the new rings in a worn liner will have you right back where you are now in short order.
As you can imagine, you are looking a very expensive repair in an overhaul like this. You had better really do your homework on selecting a mechanic/shop to help you with this or you can end up with some very pricey metal slag at the end of the exercise. There are several considerations and observations to be made at tear down to ensure that your engine block is worthy of being left in service as well. The liners come in several sizes to compensate for block wear but precise measurement must be done to fit them correctly to each hole. Because, each hole can be different from the next. Not a job for a weekend warrior or dock rat with a dock cart tool kit. Sadly, good 2 stroke Detroit techs were hard to find back when they still manufactured these engines and it has only gotten harder since.
On the up side, these engines are some of the most reliable ever built. Not only that, but they are VERY rugged. It is my experience that, if you can live/deal with the cold start issue, you can probably still get a bunch of service out of this engine if you need to put off the repair and "save up".