Yes, there are two times of year: 1.) when new models come to market and older ones get discounted for clearance, and 2.) during promotional sales periods like Black Friday.
For the first, you'll need to wait at least until March, when the new models come in and the old ones suddenly become worth less. Black Friday and Cyber Monday are little more than a month away. But you'll have to work hard to snag one of the very few real bargains. And if you miss the great deal, you'll probably have to wait for the post-Christmas sales in January.
Any time can be a good time to buy a new TV if you keep your eyes open for deals. If you're cash-strapped right now, it can't hurt to go to a retailer and say "I have this much to spend, what can you do for me?" You never know when a store or slaesperson needs that sale more than the profit.
As for getting rid of that old TV, don't try to dump your old stuff on people you know, that's just asking for hard feelings. Craigslist, local classifieds and even a bulliten bosrd in your local grocery store are good places to advertise. Ask your local police about safe places to meet. Never allow strangers into your home!
If you have no takers, try the local Goodwill or other charity. Or call your local firehouse and see if they want your old TV. If it still works well, somebody can put it to good use.