As a mortgage broker (though only in Utah), I mostly agree with Steve Ainslie. Definitely be cautious about over-committing. However, like you mentioned, it can be a real challenge to find something that fits all of the criteria in each area.
With interest rates generally trending upward but still fairly low overall, you can look at a home as a decent place to park some money. However, the interest rates are catching up with the expected rate of return. As long as you're okay with volatility, I'd continue investing as much as you can while just making normal payments. Lots of people suggest investing the money and when you have enough to pay off the balance, make the decision whether you want to pay it off or continue to let your money work for you.
In order to save money, here's what I would suggest:
1. (most important) Shop around for mortgage rates: once your credit is pulled for a home loan, it opens a 45-day window when you can have it pulled as many more times as you like and it shows up only as one inquiry. That way, you don't have to worry about your credit being hurt. I recommend getting (free) quotes from each of the following categories: online lenders, banks, credit unions, large brokerages, and small brokerages. Depending on market conditions, one or the other may have a significant advantage. I work for a small brokerage with relatively low commission, so we usually have very low rates and fees, but sometimes a credit union may beat us if you meet whatever criteria they have (place of residence, student status, an account with them, autopay, etc.). In any case, it won't hurt to shop and it could save you lots.
2. Compare your quotes: the key here is the APR. Your interest rate dictates the payment you'll make, but the APR is calculated by adding in all of the interest plus the closing costs. If the interest rate is the same, the lower APR will save you money up front.
3. Consider different terms: 30-year and 15-year fixed are the most common terms. However, there are also 20- and 25-year products out there that may give you a nice middle ground. Again though, be wary of committing to a higher payment than you can really afford.
4. Mortgage insurance: less is better. If you have the money saved for a 20% down payment, you won't have to worry about it. If it's less than that, you'll have to have it one way or the other. If you have a conventional loan, it'll automatically drop off once you've paid off a certain percentage of the home's value. If you have an FHA loan, it's there for the life of the loan. If you have poor credit, the mortgage insurance for a conventional loan may increase a lot, but FHA won't.
5. "Buying" a better interest rate: This one varies a lot from lender to lender. I'd just look at the difference between the estimated monthly payments and see how long it'd take to pay back.
I'd say those are the biggest keys. Feel free to message me if you have any questions!