$300 might be a little lean for a new saw that will make you proud. Something like a Husky 350 is a high quality consumer saw that can be had for that amount, but will have to work pretty hard for wood over 16" in diameter. If you're willing to leave the bigger stuff for someone else or take a while to cut, it might work for you. Remember that the effective cutting portion of the bar is about 2" less than the bar length, so a saw with a 18" bar is good up to about 16" in diameter.
If you are willing to spend around $450 on the net, then you can get something like a Dolmar 5100, which is one of the most talked about saws on this forum. I hear it handles a 20" bar nicely and have read a number of posts of people putting longer bars on it. I think it's really designed for 16" - 20" bars for normal use. Many will recommend you buy from a local dealer, which you may not have for this brand; it will cost you a little more but may pay dividends if things go bad.
If you really intend to cut lumber 22"-28" in diameter with any regularity, I'd recommend a 70-80cc saw. You'll have to consider a used saw to meet your price goals, but your local dealer might be able to help you out on a trade-in or a refurb. I run a Husky 372xp which performs well and is well respected on this forum. They are getting hard to find new as they have just been discontinued for emmision reasons. There are tons of used ones around and parts are plentiful. I've found this saw good for bars up to 32" when using skip chain. Baileysonline.com had a few left as of last week for $579 (must add chains and bar which might run you another $150 or so including shipping). Many of the Stihl guys like the Stihl 440 or 460 which are roughly equivalent in performance.
The Dolmar 7900 or Husky 385 would be another step up in size. My guess is this is more than you need or want for your work. If you plan on ripping big lumber as you mentioned in your previous post, it will take a lot of saw. I think you'll find splitting a better option for your budget.
If pride of ownership is a big deal, consider the Dolmar 5100 with a 20" bar and a handful of chains. Take a little longer on the big stuff or leave it for someone else. I hear its a great saw, runs around 4.0hp stock, and is light for its performance. It sounds like a lot of saw for the money. You'll have to get creative to find another $150 to buy it new, but you only have to buy it once. You'll get to enjoy it every time you use it. I'll let the guys who own one tell you which chain size to get. I enjoy the 95VP on my smaller saw, but some prefer 3/8" LG on the 5100. I imagine the 0.325" 95VP is a little easier on the saw (narrow kerf, less drag) and may be useful if you find yourself on larger wood frequently. I'll defer to the guys that actually own the 5100 to express their opinion.
Have fun and be careful.