Since TalkTalk don't own any phone lines, your broadband service will have actually been coming to you over BT Openreach's wires. Any new provider (unless you switch to a service such as Virgin Media, which has its own cables) will be using the same infrastructure as you've had in the past, meaning that your broadband service might continue to keep dropping out. A decent provider though should hassle Openreach to get the problem fixed for you. Plusnet, as recommended by Tony, is part of the same company (BT) that Openreach is, so that might be in a good position to do that hassling for you if it's needed.
Unless you live in a house with, say, lots of teenagers all streaming YouTube videos while you're also watching catchup TV over the internet, you almost certainly don't need particularly fast download speeds. The 10 Mbps ADSL 2+ service offered by Plusnet should be perfectly adequate for your needs. (I use a 10 Mbps ADSL 2+ service myself and I'm not planning on changing anything). Plusnet charge £18.95 per month but you get a £75 reward card when you sign up:
https://www.plus.net/deals/unlimited/