Unless you own, not just vassalize, the provinces they shouldn't hurt you. In fact the 'free' income should help you.
Without knowing what you're doing with the sliders, we can only make edumacated guesses as to what is going on. So, a few quick questions to help you assess:
In the budget tab, where the tech sliders are...
-Is the lowest tab run all the way left as much as you can...?
-Are you teching one item at a time or 'spreading' the wealth' to all techs at once...?
-Are you focusing on techs as you need them, or simply waiting for them...?
On the sliders tab... you should have had a number of moves by now.
-Did you go centralized to remove the penalties of decentralization...?
-Did you move sliders that penalized research...?
Have you been taking income from your budget and pocketing it...? If you have, then money that goes to tech isn't going there. As a general rule of thumb, hover over the cash on hand displayed at the top of the screen, where stability and prestige are displayed. A pop up will say you get 'x' ducats a month and 'x' ducats at the end of the year for a total of 'x' ducats. Now go into the budget tab and set the lowest slider as far left as you can so that when you close it (you must close it for any changes to take effect) and go back to the cash on hand, the pop up shows you are breaking even. If you are making 12 ducats a year, you can run at (-1) ducats a month in the budget, 72d a year and you can run (-6d) a month, follow...?
As you make more ducats, you can run say (-5) instead of (-6) and slowly build up a little nest egg. This is called 'minting', taking money out of research and putting it into your pocket. Excessive minting leads to 'inflation'. (not real world inflation, but the effects are similar) Inflation makes everything costs more, including techs.
Research one thing at a time. Early game focus is GOVT for National Ideas (NIs), PROD until level 9, for workshops, and LAND, which is obvious. TRADE is next and then NAVAL. After 9, TRADE begins to take over for PROD to get monopolies. GOVT is always a priority. The idea is you actually get to use the tech you researched to help you get other techs. It's been proven researching one at a time is faster than across the board. As you get proficient at it, you'll learn to play the sliders to maximize months and pull money out of a tech that's completing and start the next one.
Slider moves to get land/naval/trade bonuses are nice, but if you're paying through the nose in tech you're not doing yourself many favors. In the end they end up just subsidizing your tech deficiency.
Did you build a COT in your best coastal province on the mainland yet...? Very good tax bonuses, and a guaranteed income trade wise. Always a nice boost.
Anything eating your income...? Alot of troops...? Alot of ships...? Are you holding an excessive amount of men to defend because you think you can't get there in time...? Do you have an overly large Army or Navy...? Do you lower maintenance when at peace...? I personally don't, instead carrying a leaner military in general, but with the right NIs you can run 1.5 or even 1.0 on the maint slider and have more than enough to handle rebels.
Are you actively trading...? Do you have NIs that boost that...? Do you have any other financial NIs like National Bank...?
Do you keep STAB at +3 as much as possible...?
Do you have decent advisors...?
The world of EUIII finance can be a complicated place... The above is just basic stuff, a few simple ground rules that will help you survive until you get the feel of all the ins and outs of the economic system.
With more information, we can help pinpoint any specific problems unique to your game and play style.
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