What kind of cut depth are you talking? 6"? 10? In dry hardwood?
How much force are you putting on the wood when you feed it? You should only be pushing hard enough to keep giving the blade wood when it has finished devouring what it's already had - a nice smooth pressure, the blade should set the speed.
Also, what kind of fence are ya using? Pivot point? Just a flat fence like a table saw?
I don't think a bigger saw is the answer - sure is easier to spend other peoples' money, though
The tooth count is DIRECTLY related to the thickness and I don't think we have enough info to decide if 3tpi is right or wrong. And, for the record, green wood blades aren't so much about the tooth count as they are the angle those teeth are ground to. The teeth are shaped differently for green wood than dry stock.
A 1/2" 3TPI blade would be okay in 6" and taller cuts of hardwoods and should track just fine under normal circumstances.
The rule of thumb with tooth count is generally to try and keep between 12 and 24 teeth engaged in the wood at any given time. A 3tpi blade would be good between 4" and 8" board. Down in the lower tooth counts, though, you start running into some practical limitations and I try to keep it as low as possible when resawing really wide stock. I have a 2-3tpi 3/4" blade that I regularly use in 12" wide stock - ripping ~1/16" veneers from.
Resawing is very much an exercise in getting to know your saw. If your blade is wandering so much that you blow out of a 1/8" slice, then you've got to look into things like tracking and blade drift. If your fence is not perfectly aligned with the cut (not the blade!) it will probably blow out.
I don't believe anyone should settle for very much, if any, drift in their blade - I shouldn't have to tilt my fence much at all. If my blade drifts more than a fraction of a degree, then I'm not setup correctly. It's worthwhile to setup your saw for as near ZERO drift as possible. If you have crowned wheels (most of us do on smaller saws), you can very very subtly adjust the tracking on those wheels such that the blade gets twisted ever-so-slightly by the crown of those wheels to pretty much eliminate all drift. Some folks don't think it's worthwhile, i leave you to decide for yourself. I don't have drift, i know others who don't have drift, so it really can't be all that tough to get rid of.
Anywho - a few of those questions above will help us to further diagnose the problem.