Temperature for young Lovebird

We're trying as hard as we can to keep a steady temperature in our room for the lovebird we have, I believe it's 7 weeks old (maybe older I'm not sure), and the room we have is difficult to keep consistent with temperature. It never goes below 70 according to the thermostat I bought, but when the wall heater kicks on the temperature goes up to near 80 sometimes, and it's very hard to get it to not heat too much. Should I be getting a space heater that might do a better job controlling the temperature more steady? I also was worried about drafts, the best place for this bird is adjacent to the window (I noticed pretty much all the places in this room have a bit of a breeze), so I put a towel on the back of the cage and part of one side where drafts may be going. We live in California though, so any air that would be coming in never is colder than 40 degrees, and that's at night when the whole cage is covered up anyway.

I really don't want anything to happen to our precious little bird, so any advice on how to deal with our situation would be great.

If you watch your bird close if she is cold she will fluff up her feathers If she is to hot she will pant. A parrots body temp is higher than ours so if we are cold the bird wouldn't be unless ill.

Don't forget your little precious will benefit from a daily dose of direct sunlight for good health.

She appears to get chilly from time to time, she'll start closing her eyes when i'm holding her and puff up feathers slightly...i put her in her cage when she was doing this and she actually took the sleep position, i'm not sure if she just was tired and needed a little nap or if it was the temperature. Temp was closer to 70 when this happened, and when it warmed up a couple degrees she became more active again.

We should be fine if the temperature just doesn't go lower than 65 right? It's usually right around 70 maybe warmer depending on if the wall heater is on or not. The heat does radiate over toward her pretty easily. She hasn't showed any signs of being too hot though.

Right now I know it's warm enough in here, i opened the windows and the sunshine is coming in, and she is in sleep position...since she's still a baby that's probably normal for her to take little naps right?