It can be easier said than done but you must try to ignore the speed at which your opponent is playing, especially if it is a 60 minute game as you have a full hour to play your moves and should not be under time pressure for a long while. As you say, you are a beginner and are playing 60 minute games so you can think carefully and improve - you can still do this and it is still a valuable exercise even if you use 15 minutes to make 10 moves whilst your opponent bashes them out in 1 minute.
It is likely that any such player does not have your level of patience and focus which is also why they are offering draws - they do not have the discipline to take their time and think carefully and get agitated when you are not moving quickly in response. Against such players you should have a good chance to play a good game by using your time wisely and thinking more carefully than them.
Even if you are lower on time or down in material during the game it is still worth continuing as it will help you to get better and there is always the strong possibility that your opponent makes mistakes and allows you to get back into the game. I would refuse the draw offers and continue the game whilst wondering why folks with attention deficit disorder decide to play 60 minute games in the first place.
I refused the first ask and in the second ask I accepted the draw.
The pressure to play faster was too stressful for a beginner.
So my thinking is to just accept the draw on the first ask, and find an opponent who also takes more time.