You are describing a number of problems, the first thing is to break it down to each problem. BTW -- Is your TV always on CH 3 or 4 and the VCR does the tuning or do you hae both, the TV tune and the VCR? Also if you have cable from a cable company is it possible they are transitioning to digital, I know mine is and over time I have fewer and fewer analog channels. That aside...
I would start with the TV, take the cable from the wall and plug it directly to the TV and let the TV do the channel scan, see if the TV can get more stations or all. If it can then it might be your VCR that is dying.
If you use your VCR in by-pass mode, that is, when you are not using the VCR the TV tunes its own stations it could be that your newer TV requires more signal, the test above where you plug the cable directly will tell you that, if plugged in directly it gets all of the channels then you VCR is causing enough signal strength loss to make it where the TV can't tune them in.
If you always tune through your VCR the above is not a factor, your TV is alaways on channel 3/4 and the VCR is doing the tuning, if that is the case, then the issue is with the VCR, again the above test will tall you that, if your TV can get all of the stations and without snow or issue and does not blank our with a low signal message then the VCR is the issue.
As for your universal remote, you might check the manual, many TV makers have more than one IR code of their equipment and one of the symptoms of having the wrong code is that the remote will do some functions but not others. So if you have the remote manual, you might check the other codes for RCA and try them, I know RCA usues more that one code for TV's. One of the codes will likely return all functions to your remote.
Since you say it will record from the VCR fine and I am not sure how you are hooked as far as does the VCR always do the tuning or does the TV tune to stations when the VCR is off, then it is a pretty sure bet you are getting enough signal loss through the VCR to effect the TV, again the first test above with the cable to your TV directly will tell you if that is the issue.
My old 19" RCA TV (Model F19220BK) started having what I assume to be tuner problems (after the set was on a few minutes, channels 30-50 would go to snow). I have another similar RCA 19" TV (Model F19426) which I switched with the original TV set. Now I find the universal remote I'm using doesn't display the TV functions same way that it did with the older TV. I can turn it on and off, basically, but it doesn't display closed captioning or the VCR counter, and when scanning for channels doesn't add some of them. When I try to program the VCR (Zenith VRM4150) to record from the TV, every few second that the VCR is on, the TV screen goes gray and has a message "No usable signal" even though it will record from the VCR. I connected the VCR the way it was on the old TV - the coax from the cable goes to the VCR the out to the TV. What could be the problem with the interaction between the new TV and the VCR?