which amp to get?

The 1062 is a great amp. Its probably just as good as most of what is within your budget. You'll have to be careful that you don't end up with something that's not an upgrade. In my opinion, a Creek 5350 would be a very nice upgrade.

Try tubes. If I was buying an integrated amp I'd get a used Jolida or Prima Luna as they make a lot of variations of them, many with RC, and tubes are more fun.

If you can find it used look for a Jolida JD-502CRC or 502B Integrated. Some of them hover around the $650 to $800 range used. I enjoy a Jolida JD-502P amp and the integrated I mention use the same amplifier section.

given your contemporaneous post about what speakers to buy (first?), I would suggest that you decide on that first and THEN move to an amp selection, if you want to strive towards any system synergy

Avoid this random "shotgun" approach and use a "rifle" approach instead.