Be patient....if it is the right house it will work out.
The agent should be working on behalf of the seller and not themselves, but it is hard to know in this situation and in probably ever situation if it is the agent or the seller who is being slow.
Also it depends on what the contingency was. If it was financing only they may continue to work the first offer and not even respond to yours. If the contingency was to sell another house, then typically the 1st buyer has some time period to remove that contingency and move forward. If that happens, then they may offer you the chance to be a backup contract.
There's no rule about needing to talk on the phone. Most transactions can be handled by email. I really wouldn't worry about that. There's plenty of transactions that happen like this.
If you really want the house and feel like there is some ill will at this point, offer the selling agent a bonus. Give them an extra $500 of your buyer's agent commission for taking time to show the house and provide you with information. It's a good faith gesture. Discuss this with your buyer's agent and let them do the talking. Now they might not be happy either, but I would think it would not be a big deal for them either since you took very little of their time and dropped a contract in their lap from what I can tell.
It shouldn't but it can cause some ill will or bad feelings sometimes when a buyer has taken the time of the listing agent and then brings in a 3rd party buyer's agent. You buyer's agent really should be the one showing you homes if you want to use one. The seller's agent should do everything they can to sell the house, and even if it is disappointing for this to happen, it does happen now and then and they need to learn to work through it.
I'm not a lawyer, but you also might want to look up information on procuring cause. It could be that your current agent is not due a commission, even if you did not sign a buyer's rep with the first.
So that's something to think about.
Another idea....far down the list of what I think is a great idea, is that you not use your current agent, but call the broker or manager of the listing company. Explain your situation and ask that one of the agents in their office represent you in the transaction. Interview them and make sure they are the right fit for you and will really represent you. Sign a buyer's rep with them just for this one transaction. That way the broker or office manager of the company is involved, they're motivated to make both sides of the transaction happy, and maybe the deal gets done.
If it doesn't and you really like your current agent, go back to them for future showings and buys. Just be up front with everyone about what your intentions are.